The 23-year-old is believed to be under consideration from Lambert, who has identified strengthening in the middle of the park as one of his priorities during the January transfer window.
Sissoko's current contract runs out this summer and he would be available now for around £3 million.
The France international was also on the radar of the then Villa manager Gerard Houllier two years ago.
Sissoko can operate as a holding player who would offer protection to a defence which has leaked more goals - 42 - than any other Barclays Premier League team this season.
Lambert admitted last week that Villa were missing the presence in midfield of club skipper Stiliyan Petrov who is battling acute leukaemia.
Sissoko has also been linked with a move to Sunderland.
Speculation that former Villa midfielder Nigel Reo-Coker could return on a short-term basis are wide of the mark.
Reo-Coker has just ended a three-month contract with npower Championship strugglers Ipswich.