Liverpool's search for a new manager is due to end this week with an announcement expected on Friday.
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.