The Coronation Street actress is said to have been flirting with the TV presenter for months after being introduced by mutual friends, according to The Sun.
The pair were reportedly seen kissing by fellow patrons while on holiday in Dubai.
"Michelle hasn't wanted to get into anything serious since splitting with Max [George], but she really likes Mark. They both ended up in Dubai for the New Year, on separate trips, and have spent a lot of time together," a friend of the pair said.
"One night they went out for some sushi with a group of friends and things got a little frisky."
An eyewitness added: "They met in the bar and had to wait quite a while for a table. At the end of the meal Mark kissed her for five minutes in public, a big no-no in Dubai. We were shocked they were at it for so long."
Keegan ended her engagement to The Wanted singer Max George in April 2012, before they announced their eventual split in July.
Wright was recently linked with I'm a Celebrity runner-up Ashley Roberts, after they were spotted flirting at a wrap party in Australia.