The former EastEnders star was spotted with Artem Chigvintsev, 28, heading back to her north London flat on Tuesday night.
The pair - who only met 3 weeks ago - had earlier spent the whole day training together at a gym in Swiss Cottage.
'Most people involved with the show have figured out what's going on by now,' says a source.
'They've been really flirty in rehearsals. They are both young, single and attractive so it was probably only a matter of time before something happened.
'They can barely keep their hands off each other. It's sweet really.'
But Kara's rep has played down the reports.
'Of course Kara and Artem are spending lots of time together,' her spokesman tells the Daily Mirror.
'They are training every day - but they are just good friends.'
Kara, 27, split from actor Joe Swash 4 months ago after 2 years of dating.
See our Strictly gallery