Herita Ilunga's agent claims West Ham are keen to turn his client's loan switch into a permanent deal.
The Congo international joined the Hammers in September, penning an agreement which is set to run until the end of the season.
However, having impressed the Premier League outfit during his short time in England, a compromise could now be reached during the January transfer window.
The full-back is believed to have no desire to head back to Toulouse and West Ham are now investigating the possibility of handing him a full-time contract before 2nd February.
"West Ham really want to sign him, but I cannot tell you when that may happen," Ilunga's representative Daniel Striani told Le 10 Sport.
The 26-year-old has been a mainstay in Gianfranco Zola's side in recent weeks and grabbed his first goal for the Hammers during their 3-0 FA Cup win over Barnsley on Saturday.