The 22-year-old joined Thistle last summer after leaving Motherwell, where he never made his mark on the first-team, instead spending much of his career on loan in the lower leagues.
Jackie McNamara snapped him up and he was rewarded with some high-class displays and he finished as Partick's top-scorer with 13 goals.
He has 12 months left on his current deal, but there is big interest from both Scotland and England.
McNamara, now in charge of Dundee United, is understood to be monitoring the situation, but he too will be aware of the interest from England.
Crystal Palace, Millwall, Blackpool and promotion chasing Brentford are all believed to be among those keen on Lawless.