Boro had an initial bid for Wallace rejected, but they have upped their offer now and could well get a move over the line for the attacker.
Wallace, who can play on either flank or in attacking midfield, was a key player for Millwall for last season.
The 24-year-old’s energy from wide areas made him a popular figure at the Den, and he also scored seven goals.
Boro are looking to improve their attacking options, having lost Adama Traore to Wolverhampton Wanderers.
Much of Tony Pulis’ side’s attacking play went through Traore, but they are now looking at other ways to cover for the departure of the winger.
Boro are in negotiations with a number of players today, with Pulis’ side looking likely to have a busy deadline day.