The England international enjoyed a brilliant loan spell at Selhurst Park last season but has since struggled for first-team football at Stamford Bridge.
Gallagher was included in England’s World Cup, thanks largely to the performances he put in for Palace, though has been handed only ten starts across all competitions this season.
Standard Sport understands the 22-year-old is attracting loan interest from both Palace and Newcastle, although it is understood Gallagher would prefer to stay at Chelsea unless he is forced out.
Vieira is keen on signings during this month’s transfer window, particularly in attacking areas, but denied the club were yet in talks over a move for Gallagher.
"No,” he said when asked if Palace were in talks with Chelsea.
"There are options to improve that side of the game (in the market)," said Vieira. "This is a really difficult window, to find those players who can improve the squad.
"We’ve managed to identify those players, we’ve identified the names of the players we wanted, and now it’s in the hands of the football club on trying to make it happen.
"But at the same time, it’s not an easy one. It’s a really complicated and a really difficult one. We are working hard to try to bring those players."
The club are eyeing a move for Galatasaray’s Sacha Boey should they be unable to bring Manchester United defender Aaron Wan-Bissaka back to Selhurst.
Wan-Bissaka had been a loan target at Palace earlier in the window but has since enjoyed a run in United’s first-team under Erik ten Hag.