Boro have emerged as favourites to sign the 26-year-old in recent days, with the newly-relegated Championship outfit thought to be willing to offer up to £14m for the frontman.
Garry Monk's side are also prepared to pay £60,000 a week in wages to Assombalonga, who scored 14 goals to help Forest retain their Championship status last season.
However, the Nottingham Post reports that Burnley are readying an improved offer of their own having seen an initial £8m bid turned down earlier this summer.
Assombalonga is likely to favour a move to the Premier League, while Forest would also be happier to sell to a team in the division above rather than strengthening one of their Championship rivals.
Burnley could use one of their own players as a makeweight in a swap deal, while Watford also remain interested in bringing Assombalonga back to Vicarage Road following his 2013 exit.