Clichy, 31, has confirmed he will leave the Etihad this summer after not being handed a new contract by Pep Guardiola.
The France international will be available on a free transfer when his contract officially expires on June 30.
And The Sun claim Liverpool will pounce by offering him a lucrative two-year contract.
The deal will run until 2019 and apparently includes a one-year option to extend his stay at Anfield.
Liverpool transfers: Most likely summer deals Thu, April 20, 2017
Express Sport takes a look at the most likely summer signings Liverpool might secure according to the bookies. Odds provided by 888sport
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Who is Jurgen Klopp tipped to sign this summer?
Liverpool offer two-year deal to Manchester City left-back Gael Clichy.
Jurgen Klopp is desperate to sign a left-back this summer with the German unconvinced by Alberto Moreno.
James Milner has filled the void but the German boss wants to sign a specialist in that position.
And it is claimed Klopp is keen to land Clichy to resolve his defensive problem ahead of next season's return to the Champions League.
The report adds Clichy is also attracting interest from Paris Saint-Germain.
Liverpool offer Manchester City star Gael Clichy a lucrative two-year deal
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Liverpool have already got the ball rolling in the summer transfer window with Dominic Solanke agreeing to join from Chelsea.
The Reds fended off the likes of Celtic and RB Leipzig to win the race for his signature.
A fee of between £2-£3million will be decided by a tribunal.
Solanke, 19, will sign a deal worth around £20,000-a-week and will initially kick start his career in the Reds' Under-23s.