The 34-year-old’s representatives are in talks with Bournemouth, who are prepared to match his wag demands, and Crystal Palace.
West Ham were initially ready to compete with Bournenouth for the striker, who will leave relegated Sunderland at the end of this campaign on a free transfer, but have now cooled.
Defoe, who hit 18 goals this season, is asking for a minimum two-year deal from interested clubs, with an option for a third.
The Hammers have turned instead to Manchester City striker Kelechi Iheanacho as a possible alternative and are ready to battle Everton for the 20-year-old.
Jermain Defoe wants wages of more than £110,000-a-week
Slaven Bilic could turn his attentions to Kelechi Iheanacho
Everton have bid £24m for Iheanacho, but West Ham are expected to top that offer over the weekend.
Leeds’ New Zealand striker Chris Wood is wanted by West Ham manager Slaven Bilic as he revamps an underperforming strikeforce.
The Hammers are also interested in Manchester United centre-back Chris Smalling, who Jose Mourinho could be ready to sacrifice this summer as he rebuilds his squad at Old Trafford.
Premier League deals that could define the summer Fri, May 19, 2017
Express Sport takes a look at some Premier League deals that could define the summer
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Antoine Griezmann – linked with Manchester United
Sunderland boss David Moyes is resigned to Defoe’s departure but has warned Jordan Pickford’s suitors not to expect a fire-sale price after confirming the club will only consider offers of around £30million.
Everton are the latest to eye up the 23-year-old goalkeeper but Moyes said: “It would need to be a really, really big offer.
"Sometimes on your journey you may have to sell to improve, but I have heard some really derisory sort of figures getting mentioned, and it wouldn’t be any of those.”