The 20-year-old Weiss, whose father is the Slovakia manager, played at this year's World Cup in South Africa but has failed to earn a first team place at big-spending City.
"Celtic came in for me first but then Rangers came in and when I looked through all the offers that I had I picked Rangers," he told the club's website.
"The fact that Rangers are in the Champions League was obviously a big incentive for me. The fans are incredible which is another reason I chose Rangers."
"We've agreed a fee with Rapid for Nikica Jelavic and we've also agreed personal terms in principle," chief executive Martin Bain told the club website. "He'll arrive tomorrow when we'll hope to complete these and when he'll undergo a medical."
The 24-year-old Jelavic would be Rangers manager Walter Smith's third signing of the transfer window after the arrival last week of former England striker James Beattie, 32, from Stoke City on a two-year deal for an undisclosed fee.
The Croatian was pulled out of Rapid's Europa League playoff tie at home to Aston Villa on Thursday, which finished 1-1, and if the Austrian club are eliminated from the competition, would be eligible to play for Rangers in the Champions League.
