Boro boss Aitor Karanka is looking to re-work his squad and has targeted midfielder Clayton, who caught his eye last season.
Sky Sports understands Boro tabled an offer of over £1million, but Town were quick to reject the bid as they want the player to remain at the club.
The likes of Kei Kamara and Jacob Butterfield were linked as part of a player-plus-cash deal, but it is understood Boro's offer was cash only.
Clayton, who has also attracted interest from Brighton, has one year left on his current deal at the John Smith's Stadium and Town are keen for the 25-year-old to extend his stay.
Clayton, who came through the youth ranks at Manchester City, joined Town in 2012 from Leeds United.