Gibson has been on the transfer lists of a number of clubs for months.
West Brom are said to have lodged an £18million offer for the player, with £3m in add-ons.
But Boro have refused the Baggies’ club-record offer and are instead seeking a buyer from higher up the Premier League table.
That is according to the Yorkshire Post, who claim Tottenham are also in for the player.
According to the report, West Brom will have to increase their offer for Gibson to get the Boro board to the negotiating table.
Boro see themselves as promotion favourites in the Championship this season and don’t want to sell their top defender to a team who may be battling against them in the lower half of the table next term.
Yet a move to Spurs is not straightforward for the 24-year-old either.
The Yorkshire Post claim that Spurs boss Mauricio Pochettino will only bid for Gibson if Kevin Wimmer leaves the club.
A fortnight ago Wimmer, 24, said he wanted to quit the north London outfit in search of more game time.
“We will definitely look around. It’s the right time to look for a place where the chance to get regular playing time is higher,” he said.
But with few clubs willing to meet Tottenham’s asking price it appears as though the Austrian may stay this summer, casting a move for Gibson into doubt.