Albion, clear of the relegation zone on goal difference alone in 17th place, have not won any of the seven fixtures they have played under Mel since his appointment in January.
Reports have been suggesting his position is already under threat, while the Spaniard himself has made it clear he views the contest at the Liberty Stadium as a must-win game.
And asked about what might happen if the Baggies do not take maximum points on Saturday, Mel said at his pre-match press conference on Thursday: "I understand why you have to ask the question.
"But I'm sure you'll understand that I don't actually know."
He added: "My focus is on the players and working with them in order to improve them, because this is the only way I can be sure I'm going to be here for a long time."
Stretching back to before Mel's arrival at the club, West Brom have won only once in 18 league games.
They have 10 matches left to play, and while that is one more than 18th-placed Cardiff, 19th-ranked Sunderland have 12 to go and are only a point worse off than Albion and the Bluebirds.
And the 51-year-old - whose side, after Swansea, face Hull and Norwich away either side of hosting Cardiff - said of Saturday's encounter: ''Of course we must win.
''It is being said that there are four important games, but for me there is only one important thing at the moment, and that is to beat Swansea.''
Although Mel is looking for improvement in his players, certainly it seems he has confidence in them.
Asked if he could see winners in the Albion dressing room, Mel said: "If this wasn't the case - if I didn't believe there were - then I would have already spoken to the chairman in order to leave."
Mel emphasised he had not seriously thought about departing, and insisted he has no regrets over becoming West Brom boss.
With the winless run, the loss of a considerable number of players through injury and the furore over Nicolas Anelka's 'quenelle' goal celebration, it has hardly made for an enjoyable time in the Baggies hot seat so far for the former Real Betis coach.
But asked if he had any regrets about taking the job, Mel, speaking in Spanish and through a translator, said: ''No.
''I knew it wasn't going to be easy. Alongside the situation with Anelka, the injuries and of course the results, there is also the fact that it is a new country with a new language and culture.
''But, for me, it was an opportunity and a really exciting one.
''I'm a very well recognised head coach in Spain, but here I have to earn the right to be respected.
''This is what I am trying to do and the only way I know is through hard work.''
While it appears uncertain what the West Brom hierarchy think about Mel, it does, at the very least, look as if he has the backing of some of the club's supporters.
It has emerged that Albion fans are set to take Spain flags with them to Swansea as a show of support for Mel.
And asked about that, he said: ''I think the supporters are extremely important.
''Players and coaches come and go but the supporters really are the heart and soul of the club.
''I can only thank them. Since I came to West Brom, they have been extremely kind to me.
''Indeed, everyone at the club has. But what I would really like to stress is that I am extremely indebted to the supporters.
''But perhaps this isn't the best game (to display the flags) because Swansea have eight Spanish players!''
Mel also said that Steven Reid (knee), Billy Jones and Claudio Yacob (both hamstring) would not feature on Saturday.
Diego Lugano and Chris Brunt (both knee) had already been ruled out, while Anelka will serve a five-match suspension after both he and the FA declined to appeal against the sanction.