Pujara remained in England to spend a short spell with Derbyshire after India's Test series defeat, and the batsman is believed to be keen to return to the County Ground.
"We've spoken to Pujara already and he seemed quite keen to come back," Welch revealed to members of the media. "He integrated into the team brilliantly and I think he enjoyed the lads here.
"He wants to test himself in English conditions, so this is probably the best place for him."
Pujara would almost certainly miss the early stages of next season due to his Indian Premier League commitments.