Austin is out of contract at the end of the year and United boss Neil Warnock invited the player for a trial during his time in charge at QPR.
Brann have confirmed a fee was agreed between the two clubs on Sunday night and the 27-year-old is due to travel to England to discuss personal terms and undergo a medical.
"We agreed with Leeds on the conditions for a transfer on Sunday night," Brann's sporting director Rolf Magne Walstad told the club's official website.
"Negotiations have been slow, partly because the distance between what Leeds would pay for the player and what Brann wanted to get was great, and partly because we had the desire to keep Rudy at Brann for as long as possible.
"Since this interest became known it was clear that Rudy wanted to continue his career in England if the clubs agreed a transfer."