It is understood he was one of a group of five swimmers who got into difficulty in the sea at Tullan Strand near Bundoran on Tuesday evening.
Four of the group were able to make it to safety with the help of local surfers.
Bundoran RNLI have advised people to 'stay away' as the search resumes.
The Sligo-based coastguard helicopter, Killybegs Coast Guard, defence forces and GardaÃ­ are also assisting in the rescue.
Shane Smith of Bundoran lifeboat said the operation began again at 07:00 BST on Thursday.
"While the search is ongoing, we're just advising people to stay away from the Tullan Strand area for the couple of days that we are searching," he said.
"There is a lot of army personnel and shore crew up there at the minute so its just for their own safety and the safety of the crew as well."
Tullan Strand is a popular surfing location and is regarded locally as a safe place to swim.
The alarm for the missing man was raised on Tuesday by a woman watching events unfold from a car park.
The search for a swimmer who went missing off the coast of Donegal in the Republic of Ireland has entered its third day.