The striker is a free agent following a spell in Norway with Sarpsborg 08 and had been linked with a possible return to Hearts.
However, a move to Wearside appears more likely, with Sunderland manager Phil Parkinson keen to strengthen his striking options as he bids to revitalise his side’s promotion push. They are currently ninth in Sky Bet League One, nine points adrift of leaders Wycombe Wanderers.
Lafferty is understood to have met with club representatives at the Sunderland training ground on Tuesday.
The Northern Ireland international said this week that rejoining Hears would be his preferred option. Lafferty, 32, scored 20 goals in 48 competitive appearances for the Tynecastle side in just over a season from 2017 to 2018.
“My first option would be Hearts and I’ve said previously that I’d jump at the chance to go back,” he was quoted as saying in the Daily Record.
“I loved the time I had there, the fans were great with me and I have lots of friends there who I still keep in touch with. I’ve spoken with a few Hearts boys and had a few texts about what’s going on but everything is hypothetical as there has been no firm contact.
“Playing in Norway allowed me to get back out onto the pitch and get game time, it was great to be playing regularly again. Now I need to move my career on and it’s the January window so I [am] ready for whatever challenge comes my way.”
Hearts sold Lafferty to Rangers in August 2018 but the centre-forward struggled for game time at Ibrox and was released from his contract a year later. He moved to Norway where he helped Sarpsborg retain their top-flight status.
Lafferty’s peripatetic career has also seen him play in Switzerland, Italy and Turkey as well as spells in England with Burnley, Norwich City and Birmingham City.