Swann, 34, missed England's tour of New Zealand in March with the injury, but hopes to be fit in time for the first Test at Lords on May 16.
Swann is expected to play his first match of the season for Nottinghamshire against Durham in the County Championship next week, and could also play for England Lions against New Zealand in a warm-up match.
"The elbow feels great," Swann told ESPN. "I've been bowling with it over the last two days in the nets and it has never felt better.
"You go through a mixture of emotions. You have the panic, then the optimism, then the odd pessimistic day pops its head out of nowhere. But the surgery wasn't as serious as last time and my rehab has gone perfectly.
"The aim was always for me to come back on Monday against Durham and that is what is going to happen. I was just bored with stupid bowling in the nets and wanted to get back in the game."
Swann has amassed 212 wickets in 50 Test matches for England.