Tremmel, 35, is being given an extended run in the Swans first team because of an injury to No 1 Michel Vorm.
Vorm needs an operation on his knee and could be out for up to two months.
Boss Michael Laudrup has just two 18-year-old keepers, Oliver Davies and Gregor Zabret, as back-up to Tremmel.
And the German insists he is ready to scrap for the No 1 shirt with the experienced Given, if he arrives at the Liberty Stadium from Middlesbrough.
Tremmel told the Sun: "I welcome any competition. With Michel out we are very tight, although we have two young talented keepers.
"I'm always happy when I'm playing and always try my best. I just want to give the team as much as possible - but, of course, I have one eye on myself.
"It's even better when I have a few games in a row because you can get into a rhythm.
"Everyone wants to get confidence so you can show your best on the pitch."
Tremmel did his chances of keeping the shirt no harm with an outstanding display in the 1-0 defeat to St Gallen in the Europa League on Thursday night.
He pulled off a string of fine saves - including one from a penalty - to frustrate their Swiss hosts until conceding a late winner.
But he says the Swans need to step up their form - quickly - in the Premier League.
Tremmel added: "I was happy I kept the team in the game. But our performance was poor - we can't continue like that.
"The Premier League is different and I don't fear that we will have a performance like that.
"We want to make sure we are in a position where we are quite comfortable and have nothing to do with the bottom three."