Wheater is out of contract at the Riverside at the end of the season but Boro are still keen to secure the highest price possible for the 23-year-old defender.
It is believed that Bolton have already agreed personal terms with the player and Wheater is keen to make the switch to the Reebok, but the deal now hangs in the balance due to Boro's demands.
The Riverside outfit are said to be keen to sell Wheater for £4million, with the club hoping to receive £2.5million up-front.
The Wanderers bid is thought to be close to what Boro are asking for but boss Owen Coyle could still be heading for disappointment with Everton and Stoke ready to pounce if a move for Wheater stalls.
However, Middlesbrough are keen to secure a fee for Wheater rather than see him leave for nothing in the summer and Bolton could still win the race for his signature without raising their bid if no one matches Boro's £4m asking price.
Coyle has also been linked with Argentine centre-half Santiago Gentiletti, of Argentinos Juniors, and the Wanderers boss could step up his interest if a bid for Wheater fails.
There is still speculation that Gary Cahill could be sold in January, with Arsenal and Chelsea heavily linked, and although Coyle is eager to retain the England international defender the decision could be taken out of his hands if a big offer is made.
If Cahill does leave Coyle will need reinforcements and should a move for Wheater fail to materialise he may step up his interest in Gentiletti, who has also been linked with La Liga outfit Espanyol.