Spurs boss Harry Redknapp had predicted a number of his younger and fringe players would temporarily leave the club after their Europa League exit.
Livermore has indicated an openness but says his priority is to force his way into the first team at White Hart Lane.
“I'll keep my options open for January and see where it leaves us," he said, according to The Sun.
“Would I consider a loan? If it's the right club for me and they play a certain style, yes.
“Ultimately, it is down to myself and the gaffer. But I'll just keep in his face and push on from there.”
The 22-year-old was part of the side that won 4-0 at Shamrock Rovers last week, but Spurs were still eliminated due to results elsewhere.
Redknapp had been using the competition to blood his youngsters and appease his fringe players with more playing time.
This selection policy backfired after a series of disappointing results saw the White Hart Lane club knocked out by PAOL Salonika and Rubin Kazan.
Livermore featured in every one of Tottenham’s Europa League games this season and has made 19 appearances altogether this season.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms