Steve Bruce’s team are three points off the relegation zone in 15th place, but five of their last eight matches pit them against clubs in the top seven in the table.
Some of Hull’s most impressive performances have come against top sides, however, drawing 1-1 away at Manchester City and narrowly losing out to Chelsea when they could have taken something from the game.
Ahead of the weekend meeting with Swansea, Meyler said: “We are not going down. We just have to get into the mentality that it is not good enough to accept the 3-2 defeat to Chelsea, even if they are likely Premier League champions.
“We had some great chances and we need to win these games when you get into those positions. We’re down to the last eight games now and this is where we have to earn our money.
“This is the period when everything will be decided so we’ve got eight games to make sure we finish as well as we can and that is what we aim to do.
“We finished 17th last season and at the moment, we’re 15th this season so we’re a little bit ahead.
“What we want to do here is maintain Premier League status for a number of years to come and we’ll do what we have to do to achieve that aim and build this club. We’re looking to develop every year.
“We’ve got eight games to come, against some decent teams, but I’m sure we can get the wins we need if we keep performing like we did in games against the likes of Manchester City, Arsenal and Liverpool and for 70 minutes against Chelsea.
“It will be tough against Swansea but we’re going there with the mentality that we can get some points.”
Hull will be boosted by the return of Senegal midfielder Mohamed Diame after a long injury absence, and James Chester is also back in the squad to face Swansea.
Gaston Ramirez is fit to play despite concerns about an injury picked up on international duty.