If they do, they will be met with a flat refusal from Celtic, who have no intention of allowing the 25 year-old keeper to move just a year after they signed him from Newcastle for £2million.
Reports from Lisbon claim Benfica have the giant Geordie on a wishlist, and that he has moved up a place after attempts to sign Julio Cesar from QPR collapsed.
However, the Portuguese club are in no position to make big-money bids for any more players until they offload some of the men they already have on the payroll.
And with the transfer window entering its final week, there is very little time to get the cash they need to come in for Forster.
They have had no encouragement from Parkhead, and Neil Lennon made it clear after Kelvin Wilson was allowed to leave to rejoin Nottingham Forest there would be no more key players sold in this window.
That position remains the same, irrespective of whether tomorrow night's play-off tie against Shakhter Karagandy sends the Hoops into the group stage of the Champions League or the Europa League.
Lennon is only interested in strengthening his squad, and Schalke striker Teemu Pukki could be on his way to Parkhead after the Bundesliga side's Champions League play-off against PAOK Salonika tonight.
Celtic are also pursuing other targets, including another striker, a midfielder and cover for left-back.
But any hopes of landing Icelandic international Alfred Finnbogason took another hit when Cardiff entered the race for the Heerenveen striker.
That's bad news for Celtic, who baulked at Heerenveen's 5m Euro asking price for the 24-year-old.
The cash-strapped Dutch club turned down an offer of 4.5m Euros from German side Augsburg and hope to raise the fee by having as many clubs as possible battle it out for Finnbogason.
Celtic retain other options and Pukki now looks to be firmly established at the top of this list.
The Hoops are also reported to be tracking Israeli midfielder Nir Biton, whose agent is Dudu Dahan, the man who brought Beram Kayal, Efe Ambrose and Rami Gershon to Parkhead.