The former Manchester United youngster has been in training with Rovers since the start of pre-season and featured in the first three pre-season friendlies, scoring five goals.
And his performances have convinced Wellens to offer a deal.
“A contract offer has gone out to Aidan,” the Rovers boss said after the 3-2 defeat to Newcastle United.
Barlow did not feature in the friendly due to illness but Wellens expects there to be no snags in getting the deal done.
He said: “Aidan woke up, like most of us in the last few days because the weather has been unbelievable, with a headache and didn’t feel that he was fit to play.”
Barlow, 21, has been a free agent since leaving Manchester United last summer.
He did not make a senior appearance for the Red Devils but had a loan spell with Norwegian side Tromso.