But Warburton quickly denied handing in his resignation.
But King claims he was approached by their agent, who offered their resignation with "immediate effect without compensation" as long as Rangers "agreed to waive compensation from any new club that they signed for".
Warburton had been linked with the Nottingham Forest job but the Championship side decided to retain their interim team of Gary Brazil and Jack Lester until the end of the season.
Mark Warburton has left as Rangers manager
King wrote: "There were rumours that the management team (presumably their agent) was negotiating with English clubs and, in one instance, I was informally approached to ask if the Club would waive compensation if the management team was to leave. While this was unsubstantiated by direct confirmation from the Club in question, I was alert to a conversation that Mark Warburton had with me after joining the Club in which he advised me that his long-term ambition was to manage in the EPL and he viewed Rangers as a stepping-stone to achieve this. His comments to the media simultaneously reinforced his present unhappiness at the Club.
"I was therefore not surprised when the management team’s agent approached the Club’s Managing Director Stewart Robertson to request a meeting which was held in Glasgow on Monday this week. The outcome of this meeting was that the agent subsequently offered that Mark, David and Frank would resign with immediate effect without compensation as long as the Club, in turn, agreed to waive compensation from any new Club that they signed for. After discussion the Board accepted this offer and employment was immediately terminated. In order for us to achieve our ambitions we need employees that, like your Board members, will always put Rangers first.
"While we were dealing with the admin and press releases relating to the resignation the agent again contacted us and asked to defer the resignation until the management had secured a new club. I assume that the new deal had somehow collapsed at the last minute. The Board met to consider this request but resolved to hold them to the original agreement.
"We are now in the process of reviewing the best interim and long-term solution for ensuring that a modern and robust footballing structure is put in place that will continue with and entrench the footballing philosophy that we have in place. We also must protect and support the marvellous work that has been achieved by the Academy over the last two years."
Warburton has been linked with the Nottingham Forest job
Express Sport understands part of Warburton’s breakdown with the Rangers board surrounds the club’s transfer policy.
And King has set out a four-point plan to what he believes will bring success to the club, including details of their transfer policy.
The statement read: “For the avoidance of doubt, I repeat the key elements;
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“1. We would invest sufficient resources to ensure immediate promotion to the SPFL.
“2. In season two we would further invest to be competitive in the SPFL and qualify for Europe at the end of the season. Our realistic expectation was to come second. This was to be achieved by signing five or six players of a quality that improved the squad that won the Championship.
“3. In season three we would again invest in five or six players that further improved the squad to compete for the title and progress in Europe.
“4. I personally estimated that we would require an investment of £30 million over that period to achieve our stated objectives.
Warburton guided Rangers to promotion last season
“I now comment on each of these elements;
“1. We hired, at short notice, a relatively untested management team that recruited a number of players and introduced a style of play that was pleasing to our supporters. Significant investment was made on and off the park and the Championship was ultimately won in some style. The season was an unqualified success and the management team was rewarded with a vastly improved contract.
“2. This season we did not stick to our plan of signing five or six players because the manager appealed to the Board for additional signings. Despite the concern about departing from our plan of prudent phased investment, the Board backed the manager’s request for accelerated investment. This placed us significantly above the football resources available to our competitors (other than Celtic) and was expected to ensure that we finished a strong second in the league and had a squad that could be added to, close season, to make a strong impact in the Europa League qualifiers. While I still believe that we can finish a strong second, I am stating the obvious to admit that we are not where we anticipated we would be at this stage of the season and we have not repeated the success that we had with our signings from the previous season.
“3. Following from the above it is clear that we are behind our target for next season but, given that we recognise this, it is the duty of the Board to take steps to get things back on track. That is what our supporters trust us to do and rightly demand that we do. We remain 100% committed to the plan we commenced with and that the supporters continue to endorse resoundingly.
“4. £18 million of the originally estimated £30 million investment has already been made. Ultimately, the overall investment in any football team is driven by the net player spend and, given that we are behind target with our squad, there may be a further need to accelerate investment at the end of this season. It is my present personal view that we will, in all likelihood, invest more than £30 million before we are where we want to be but this will be revisited once we have a new permanent management team in place."