The Daily Mail says Tottenham have been told they will have to increase their offer of £8million for Downing if they want to complete the deal in the next 24 hours.
Boro chairman Steve Gibson is understood to want a fee closer to £10m.
Downing confided in close friends before Boro's game at home to Liverpool that he expected it to be his last for the club and waved to all four stands as he left the pitch on Saturday.
His agent Ian Elliott is leaving the country tomorrow and wants the transfer to be completed before then.
Everton are also interested in the England international but it is believed the winger prefers a move to London.