The 32-year-old former Red Bull Salzburg stopper has been on trial with the Swans, and impressed in their pre-season friendlies against Celtic and Real Betis.
Rodgers has already added two goalkeepers to his squad during the summer, with Michel Vorm and Jose Moreira arriving in the wake of the departure of Dorus de Vries to Wolves.
But the Northern Irishman is keen to add another, so he can allow youngster David Cornell, who has joined Hereford on loan, to get first-team experience elsewhere.
He said: "Gerhard has had his medical, he has gone through a lengthy trial period with us and he will be a terrific support 'keeper.
"I was desperate to get David Cornell out on loan, and thankfully we have now done that.
"We have our number one 'keeper in Michel Vorm, who has proved to be fantastic signing, and we will have two back-up 'keepers, hopefully Gerhard and Jose Moreira. Hopefully we can get a deal for Gerhard tied up in the next day or so."
Rodgers also revealed that Dutch midfielder Ferrie Bodde, who has been bedevilled by knee injuries in recent seasons, has had a spell in Holland after experiencing some discomfort in his recovery from his most recent problem.
He said: "Ferrie had to go back to Holland, he had a little niggle in his knee, he went for a scan and he is feeling more confident now.
"He had a little work done over there and he needs to rest for three or four days before he comes back to us. I didn't want to rush him and I want to give him every chance to complete his recovery so he went back to the specialist who did his operation."
Source: PA
Source: PA