Both clubs were offered the chance to sign the Portugal midfielder last week with Spurs coming closest to nabbing him on deadline day.
Gomes favoured a move to Tottenham to work with Mauricio Pochettino and was ready to join them on loan for the rest of the season.
But the move fell through when Barcelona pulled the plug at the last minute after failing to sign a replacement in time.
Gomes only joined Barca from Valencia in a transfer worth up to £50m last summer but has barely featured since, often being forced to play out of position.
The 24-year-old’s future remains up in the air but Spurs believe he will be made available again when the transfer window reopens.
Pochettino is still looking to boost his squad despite bringing in Fernando Llorente, Serge Aurier, Davinson Sanchez, Juan Foyth and Paulo Gazzaniga.
And Gomes, who won Euro 2016 with Portugal, will remain on his radar over the next few months with a view to a January move.
West Ham are also monitoring developments but Tottenham are currently in pole position should Gomes be allowed to leave Spain.
Meanwhile, Spurs have scotched suggestions Erik Lamela’s troublesome hip injury could put his career in doubt.
Villa said: “There is concern to know if he will play again. In England they are worried about him. I hope he can come back.”
But club insiders insist Lamela, who was left out of their Champions League squad, is in the final stages of his rehab and poised to make a full recovery.
Lamela, a £30m signing from Roma in 2013, is back in training at Spurs but remains “several weeks away” from a return to action.
