Peter Ridsdale, advisor to North End owner Trevor Hemmings, was present to see the 24-year-old net his first international goal in the Republic of Ireland’s 2-1 win over the USA in Dublin on Saturday night.
The former Aston VIlla youngster, who has also had a spell with Notts County, has 10 goals to his name so far this season including four in a 6-1 win over Derry City in March.
The Irish Sun reports PNE could table a bid of around Â£200,000.
Any move would see North End head into familiar territory with Sean Maguire, Kevin O’Connor, Daryl Horgan and Andy Boyle having all been recruited from the League of Ireland in the past couple of seasons.
Burke made his international debut in the defeat to France in Paris last Monday before being handed a start at the Aviva Stadium, finding the back of the net with a close-range finish just before the hour mark as Martin O'Neill's side came behind for victory.
Horgan then took the striker’s place for the final half an hour as he won his third cap.