The 29-year-old visited the club's £10.6m complex on Thursday, as the Clarets looked to thrash out a deal with his club Wolfsburg and the player, ahead of Monday's transfer deadline.
The club remain confident of landing Weghorst in a £12m deal, as well as Croatian international wideman Mislav Orsic, who, like Weghorst, is eager to try his luck in the Premier League.
Weghorst trained individually back at Wolfsburg today, as did Josuha Guilavogui and Daniel Ginczek, who are both linked with moves to Bordeaux and Fortuna Düsseldorf respectively.
Wolfsburg have no game this weekend, but Orsic's club Dinamo Zagreb do, a top of the table clash with Rijeka, which Dinamo are eager for Orsic to feature in.
A deal is still being negotiated with Dinamo, with supporters also keen for Orsic to stay, taking to the streets with banners over the last couple of nights.
Potentially, the pair could come in at around £19m, leaving Burnley £6m change after Newcastle triggered Chris Wood's £25m release clause, and, on the face of it, that would amount to remarkably shrewd business in the circumstances, if they can get both deals over the line.
Burnley wanted a midfielder of the ball manipulator type, as well as a wide player and striker, and they are still on the hunt for the former, with an interest in Juventus' Wales international Aaron Ramsey, who looks certain to leave Turin, initially on loan.
Crystal Palace are believed to be Ramsey's preferred destination, but Patrick Vieira has his eye on Manchester United's Donny van de Beek on loan, and if a deal for the Dutchman goes through, Ramsey's options appear to be diminishing.
Meanwhile, striker Andy Carroll, who was an option, but some way down Burnley's list, has linked up with West Brom on a free transfer after his short-term deal with Reading expired.