Finnigan is out of contract on May 31 and the Imps have decided to forego a transfer fee in favour of saving money - in excess of £20,000 - on the Sincil Bank wage bill.
The former skipper explained the move – "I got a call from the gaffer telling me Cheltenham wanted to take me until the end of the season and he said he would not stand in my way if I wanted to go," said Finnigan, who will be 26 later this month.
"I weighed up my options and at the end of the day Cheltenham have a great chance of promotion and if I impress down there and fit in, then I could be playing in Division Two next season.
"It is only a short-term deal but it suits me because I can still keep my options open when my contract runs out in the summer.
"It has all happened very quickly but I am looking forward to the new challenge, although it has been a wrench leaving Lincoln.”
Imps boss Alan Buckley was loath to lose his skipper but he understood the depth of City's current financial plight left him with little option.
"Sometimes you have to let the player make a decision and I understand why Finns has gone, although he will be greatly missed," said Buckley.
"He is a good player. He was a good captain and a key part of the side. But when you look at it, he has taken the chance to leave us and where we are in the League for a place in the top seven with a club which could be playing in the Second Division next season.
"I bear him no ill feeling and wish him all the luck in the world. It's not as if we will miss his goals!"