A run of four league matches without a win has seen the Welsh club slip towards the bottom three, with 18th-placed Sunderland - who have two games in hand - just four points behind.
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And Williams confessed that his side are very much in the relegation dogfight.
"As every game goes on that we don't win I think it's a test of character and we'll see in training now who really wants it, who really wants to fight," the defender told Wales Online.
"It's about the standard we've set ourselves. Over the last few years we've worked so hard to build a team and a squad. The way we go about our business has been so good and we've had a lot of praise.
"This season I don't think we've acquitted ourselves as well in the league as we have done in previous seasons. But it's not true that we're too good to go down. Anyone can go down and we've seen good teams go down before.
"We're not looking at going down, but we're in a battle and that might be a bit foreign to a few of our lads. We'll see what it's all about, we'll see how much bottle we've got."
Swansea travel to Everton this weekend to face former manager Roberto Martinez, with Monk's men looking to claim a first away league win since November.