The Cameroon international will be out of contract at the end of the season and is expected to quit Tynecastle.
Hearts had opened talks over a new deal but Djoum’s demands appear to be out of their reach, and he is now on a list of summer targets for the French Ligue 1 side.
Toulouse are currently in the bottom half of the table and coach Alain Casanova is looking to rebuild ahead of the new season.
Fellow top-flight side Dijon are also interested but they are in a relegation fight and it would depend on them beating the drop.
Djoum knocked back a lucrative move to Saudi Arabia earlier in the campaign.
The 29-year-old joined Hearts from Polish side Lech Poznan in 2015.
The former Roda player started out in the Netherlands before moving to Turkish side Akhisar Belediyespor in 2014.
Djoum has played 32 games for Hearts this season, plus a clash for Cameroon against Brazil back in November.
The midfielder has scored just three goals, picking up 10 yellows and a red.
Meanwhile, Hearts boss Craig Levein admits the team’s season run-in will be difficult.
“Yes it’s tough, but it’s good. I was disappointed in our play against Hamilton when a team sits in and we have to make a lot of the running. It’s different when we are playing teams who are going to have a go. That’s a different thing altogether,” he said.
“I think it could suit us better. We have a lot of exciting games coming up.
“There are two Edinburgh derbies still to play.
“We’re still aiming to finish as high up the league as we possibly can. There is loads of stuff to look forward to.
“I’m feeling quite good about the next phase and I think the international break came at a good time.
“We have had a few boys away on international duty but they will be back and ready to go come the weekend.”
