Liverpool's search for a new manager is due to end this week with an announcement expected on Friday.
Wigan's Roberto Martinez had been the front-runner for a long time, only for Swansea's Brendan Rodgers to take over as the bookmakers' favourite overnight.
The pair have been the two main candidates throughout the two-week process of whittling down potential contenders and Press Association Sport understands an announcement will be made on Friday.
Both Martinez and Rodgers returned to Britain in the last 24 hours after trips across the Atlantic but only the former had spoken to Reds principal owner John Henry, interrupting a holiday in Barbados to fly to Miami for brief talks.
Rodgers, along with his wife, was a guest of the Football Association of Wales for Sunday's friendly against Mexico in the MetLife Stadium in New Jersey. The Northern Irishman met with chairman Huw Jenkins on Tuesday afternoon but the club insist it was a scheduled appointment and he had not cut short his trip to America especially for it.
Swansea stress they have received no second approach from Liverpool - Rodgers having turned down their initial advances almost a fortnight ago - and Press Association Sport understands the pair did not even discuss the Anfield vacancy at their meeting.
The primary topic of conversation appears to have been transfers, with the Swans on the verge of completing a club-record £6.8million transfer for on-loan Hoffenheim midfielder Gylfi Sigurdsson and interested in a loan deal for Tottenham's out-of-favour Giovani dos Santos - whom Rodgers watched in New York.
However, despite suggestions elsewhere that Rodgers had already been offered the Liverpool job, the club refused to comment officially on their manager's position.
On Monday, Jenkins dismissed the Reds' renewed interest as "pure speculation" and the club issued a statement to stress their position had not changed from two weeks ago when Rodgers turned down the chance to speak to Liverpool.
The statement did little to quash speculation and overnight bookmakers moved Rodgers ahead of Martinez, who flew into Manchester Airport on Tuesday morning from the Caribbean to be confronted by a television camera crew and photographers but left without commenting.