The Swans have lost three out of their six games this season and have picked up just five points since Paulo Sousa replaced Roberto Martinez as manager.
After a 2-0 defeat to Preston on Saturday, Tate claims the Portuguese coach left his players in no doubt as to what will happen if they do not turn their form around.
"Paulo said there's another game Tuesday but he's going to have a look at it and if you don't do what he says then you're out the door, which is fair enough really," Tate told the Swansea Evening Post.
"We all knew it was going to be harder this season because teams realise the way we play.
"We've just got to stick at it because you can't throw the towel in if things get a bit tough."