But after enjoying little progress over the weekend it has been looking less likely that he will meet his target.
It is understood it could even take "a few more days" to make an official announcement with "two or three" contenders still in the running.
Former assistant manager Graeme Jones has been one of the other frontrunners for the job.
Jones, 40, left the Swans last summer to continue as Roberto Martinez's assistant when the Spaniard took charge at Wigan Athletic.
He enjoyed winning promotion from League One in 2008 as champions and, in their first season back at Championship level, the Swans flirted with the play-offs and was known to be a popular figure around the Liberty Stadium.
Jenkins, who lost boss Paulo Sousa to Leicester last week after just a year in charge, said on Saturday that any talk that Jones had already been offered the job was "complete lies".
Jones was in Wales at the weekend as part of a pre-planned visit to south Wales.
Pre-season training is being taken by coaches Alan Curtis and Colin Pascoe until Sousa's successor is found.
The Swans are due to travel to Hereford for a friendly on Tuesday which will be followed by the short trip to Neath two days later.
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