Meanwhile, summer arrival Micky van de Ven has been out since November with his own hamstring issue and is not expected back until mid January.
As a result of their fitness issues at the back, the Daily Mail reports that Spurs will 'prioritise defensive reinforcements when the transfer market opens', with the club interested in Liverpool's Gomez, as well as Jean-Clair Todibo (Nice), Lloyd Kelly (Bournemouth) and Radu Dragusin (Genoa).
Tottenham manager Ange Postecoglou has admitted he would like his club to get their January business done early with the window re-opening on Monday.
"Obviously if we can do business early it's great," he said. "Not just because of losing Romero, but there's a whole month there - why waste it?
"Even if they don't play, you can bed them into training and our style of football, because it's not like we're going to sign somebody and they'll hit the ground running.
"Whereas if you leave it towards the end of January, it's potentially not until the middle or end of February when they get up to speed, depending on the league they're coming from.
"I'd love to do something early, but even me saying that puts a challenge on us. Other clubs know we want to do something early, but I'm hopeful it will be much earlier than the end of the window before we bring someone in."
Any interest in Gomez is more than likely to be knocked back by Liverpool, with the 26-year-old now more important than ever to Jurgen Klopp's side amid the Reds' own injury crisis.
With Kostas Tsimikas suffering a broken collarbone and Andy Robertson not expected back until the end of January as he continues his recovery from shoulder surgery, Gomez is now Liverpool only senior option at left back.
The former Charlton star – who is currently the Reds' longest serving first-team player, having signed back in the summer of 2015 – impressed in the 2-0 win over Burnley on Boxing Day.
Speaking before that game, Klopp joked that he would have to wrap Gomez 'in cotton wool' in order to keep him fit amid the precarious defensive situation at the club.
"You have injuries sometimes and we accept all of them, but a broken collarbone [for Tsimikas] is really bad because it just takes long," Klopp told reporters. "Same what Robbo had: a different injury but that takes long as well, how we all know.
"We all know we need luck in these moments to get through and now we will have Joey there and just have to make sure we put him in cotton wool and we have to make sure he is always ready for the games."