Konate missed the final two Liverpool games before the September international break due to injury, but was initially called up to the France squad.
Despite Didier Deschamps optimism, though, the centre-back was soon replaced by Jean-Clair Tobido, having insufficiently recovered.
He still returned to France, for a mid-season holiday in Paris, which saw him provide an update on his fitness on Saturday.
On his Instagram story, Konate shared a brief video while out on a run, suggesting he is close to being able to return to training with Liverpool.
The Reds will not report back to the AXA Training Centre until later next week, with their next game being the trip to Wolves in the Premier League on September 16.
Hopes could have increased over Konates involvement at Molineux, with Klopp having already explained that his No. 5 was expected back after the break.
Whether Konate will come straight back into the starting lineup will depend on how he responds upon rejoining training.
The absence of Virgil van Dijk could boost his chances of making the squad, however, with the captain suspended for a further game due to the reaction to his red card against Newcastle.
Joe Gomez and Joel Matip filled in for the 3-0 win over Aston Villa before the break, with Klopp full of praise for his stand-in centre-backs.
Both centre-halves played an exceptional game with different jobs, to be honest, the manager said.
They could, therefore, be retained against Wolves before normal order is resumed in the following games.