Chairman Phil Wallace said: "Michael had firm offers from two League One clubs and the club opened negotiations for compensation with both last week.
"We made him a good offer to sign a new deal in line with our pay structure and we've not moved from that position."
Bostwick signed from Ebbsfleet in July 2008 and has scored nine goals in 90 appearances since moving to Broadhall Way.
He is one of our most vital players, but with offers from League One clubs it will be hard to keep him
Liam Govey
"We believe he will continue to improve and be an important player for us next season in League Two," added Wallace.
"We're delighted he's staying, he's a great lad with a big future in the game."
Stevenage boss Graham Westley has been busy so far this pre-season adding wingers Peter Winn and Robbie Sinclair and defenders Luke Foster and Darius Charles to his squad.
Source: BBC Sport
Source: BBC Sport