But it is likely to take at least two weeks before the connections will be set up, which will allow Iranian lenders to engage with the global banking world for the first time since international financial restrictions were imposed in 2012.
A nuclear deal between world powers and Iran led to the removal of the curbs on Tehran's banking, insurance and shipping sectors at the weekend, as well as restrictions on oil exports.
The SWIFT system, or Society for the Worldwide Interbank Financial Telecommunications, is used to transmit letters of credit and payments.
"We have sent almost 40 SWIFTS to different banks around the world and we have requested that now that the sanctions are lifted, we would like to exchange documents and whether they will consider a correspondent banking relationship," said Parviz Aghili, chief executive and managing director of Tehran-based Middle East Bank.
"Some of them have come back and have asked for various questions, for documents they need."
"My feeling is it is going to take a couple of weeks or so before we start to see proper re-engagement. It will be slowly, slowly," he said in an interview.
Aghili said other Iranian banks were in the same situation regarding SWIFT as his company, which is privately owned by investors that include small and medium-sized Iranian firms.
SWIFT did not respond to a request for comment on the status of communication between Iranian and international banks on its system.
Aghili also said he anticipated Iran's central bank would at some point force lenders to adhere to international capital standards, known as Basel III - a move that would press them to bolster their balance sheets.
A senior official with Iran's central bank declined to comment when asked on Friday about possible measures.