But Black Cats boss Steve Bruce has learned that Inter Milan have given former Portsmouth midfielder Sulley Muntari permission to head for Wearside to discuss a loan move until the end of the season.
It is approaching three weeks since Sunderland first made their interest known in the pair and both deals are in danger of running all the way up to the transfer deadline on Monday.
Sunderland remain hopeful at least one, Sessegnon in particular, will be struck sooner rather the later, although it has become clear within the last 20 days that nothing should be taken for granted.
Sessegnon, who remains a target of Everton even though they would prefer to do a deal in the summer, is the deal that has been the most problematic even though it was Muntari that had shown initial reluctance to head for Sunderland.
It has emerged that Sessegnon has flown over to England within the last 48 hours to discuss the finer details of a move from Paris St Germain.
But it is understood there is still divided opinion within the PSG ranks as to whether or not the Benin international should be allowed to move to Sunderland before the transfer deadline.
PSG president Robin Leproux has outlined a desire to retain Sessegnon, having been annoyed by the way the player has gone about engineering a switch to the Premier League.
As Sunderland had been looking to set up a loan deal until the end of the season, albeit with a view to a permanent transfer, Leproux had been standing firm in the hope another bidder comes in.
Everton are being kept up to date with developments, but Sunderland know PSG coach Antoine Kombouare would prefer to let him go now, particularly as the Black Cats are pressing for a permanent arrangement.
Sessegnon has been at loggerheads with the Parc des Princes club for much of this season and he had not trained for around a month until he was called back last week. The paperwork over a work permit should not take long to ratify.
Muntari, on the other hand, and his advisors are expected to discuss a temporary switch to the Stadium of Light within the next 24 hours after Inter granted him permission to return to the Premier League.
The 26-year-old has been given an insight into what life is like at Sunderland by compatriots Asamoah Gyan and John Mensah and is ready to take up the offer even though he would have welcomed a move to London.
Muntari was left out of last night's squad for the Coppa Italia quarter-final with Napoli and Sunderland are hoping to have him on board along with Sessegnon for next Tuesday's visit of Chelsea.
The likelihood is that Sunderland will still have a new striker on board by that time as well, although Bruce is adamant he will not be forced in to paying over the odds.
He has inquired about Ricardo Fuller and Jay Bothroyd but was put off by wage demands and the asking price from Stoke and Cardiff respectively.
He has also been credited with an interested in West Ham's Frederic Piquionne, but is unlikely to spend £4m on a 32-year-old.