Palace have not won a match in 2023 and were beaten 1-0 by Brighton on Wednesday, their third defeat in a row.
They are 12th in the Premier League but are only three points above 18th-placed Bournemouth, who have a game in hand.
Frenchman Vieira was appointed before the start of last season and led the Eagles to a 12th-placed finish, earning praise for an attacking style.
However, this season they have scored just 21 goals in 27 games - only Wolves, Everton and bottom-of-the-table Southampton have fewer.
Before their defeat by Brighton, Palace had gone three consecutive games without a shot on target.