The League of Ireland club have been in touch with North End about taking Burke back to Tallaght for the remainder of the season.
As things stand, there has been no agreement struck with regards to the contribution Rovers would make to Burke's wages.
Burke joined North End from Rovers in June last year for a fee of around £250,000.
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Although he started last season in the first-team picture, the 25-year-old dropped out of favour until December when he returned to feature in eight successive games.
At the end of January Burke was loaned out to League One side Gillingham where he made 12 starts and scored once.
He has featured in all but one of North End's pre-season friendlies this summer, scoring in the 2-0 win at Chorley.
PNE are slightly surprised that the only interest has so far come from Ireland.
If Burke was to go back to Shamrock Rovers on loan, their season finishes in the autumn and there would be a gap when he couldn't then sign for anyone else until the January window.
Speaking recently about Burke, Preston boss Alex Neil said: "The thing with Graham is that his best position is as a No.10 and he has competition from Alan Browne and Brad Potts.
“In the last two seasons Alan has scored a lot of goals for us playing there so really it’s difficult for Graham.
“He needs to keep performing as best he can and go from there."