Forster is among a list of top earners they want to offload and allowing him to return to Celtic last summer was an attempt to put the keeper in the shop window.
And now Lennon will have to convince the 32-year-old his future remains in Glasgow after a top-four Premier League side placed Forster on a shortlist of four keepers they are looking to sign.
Southampton are waiting to see what offers they receive when the transfer window re-opens.
The Saints are also keen to bring in as much as they can for the Geordie, who they paid Celtic £10 million to sign back in 2014.
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