The 25-year-old star enjoyed a few drinks with the 22-year-old singer-songwriter at an East London pub before they headed to Groucho Club in Soho, London, where the pair were was busy chatting and holding hands following the Q Awards at London's Grosvenor House.
A source told The Sun newspaper: ''They looked very chummy.
''Eliza kept playing with her hair and at one point they started holding hands.
''They looked like they were having fun, Tom had her laughing her head off.
''Their mates just left them to it as they chatted. Once they were done they nipped off together, but Eliza came back to the party alone.''
Eliza has remained single since her split from Good Charlotte rocker Benji Madden in 2012 after a year-long romance.
Earlier this year Tom was spotted with Taylor Swift at the same club after meeting the country singer at the BRIT Awards.
He is being tipped for big things in the music industry this year and he has previously admitted he has been enjoying all the female attention he has received at his gigs.
He said: ''I think I'm single. There are a lot of women at my gigs. The first show I headlined was a sea of women, which I can't complain about. I'm pleased these 18-year-old girls like my music. How will I make the most of it? I don't know. It's all quite new.''
The '12 Years A Slave' director will receive the accolade at the London Film Festival in October.
Critics from all over the world were asked to name the best movie of the past 16 years.