The Portuguese player, who quit Tynecastle last month at the end of his contract, has surprised the Ibrox club with his wage demands.
Goncalves is reported to be looking for a take-home wage of 18,000 per week after tax. That would mean Rangers paying a gross wage of 29,000 per week for the 24-year-old – too much for the SPL champions as they attempt to operate within a strict budget.
Manager Walter Smith expects teenager Danny Wilson to leave Ibrox with Liverpool believed to be ready to pay 2.5million rising to 4m after appearances.
Madjid Bougherra has also attracted attention with his displays for Algeria in the World Cup Finals, particularly his performance against England in the goalless draw on Friday.
Smith has been searching for replacements and the versatile Goncalves was on his radar.
Hearts made an improved offer to the Lisbon-born stopper who was picking up around 12,000 per week with the Edinburgh side. But he has elected to leave the club and secure a future elsewhere.
Rangers fans admitted yesterday that they are "bracing themselves for the worst" after Bougherra's eye-catching display against England.
That is the view of John Macmillan, secretary of the Rangers Supporters Clubs.
Speculation has surrounded Bougherra's future for months and he now appears to have boosted his chances of sealing a move away from Ibrox this summer, according to the fan clubs' representative.
"He's a good player and whether we will be able to hold on to him is another matter now," admitted Macmillan. "If Algeria go any further in the World Cup, it will be a big stage for him. I think he will be a very difficult one to hold on to.
"We are probably bracing ourselves for the worst and an offer will come for him to go to the Premiership or, certainly, one of the top Championship clubs."
Reports today have linked Bougherra with a 6m move to Arsenal as a result of his efforts in South Africa so far.
Macmillan is not sure the Algerian is ready for a switch to the Gunners but believes a mid-table side in England's top-flight is a more realistic option.
He said: "He's an excellent player but I couldn't see him at Manchester United or Chelsea, simply because they have an abundance of stars there, not that Arsenal haven't.
"I would think it's more likely he would go to one of the lesser sides in the Premiership.
"However, I don't know if he could fulfil any real ambition by going to teams, with all due respect, like Wigan or Bolton or other clubs of that ilk. It's one of the top leagues in the world so, regardless of who you are playing for, you are in among some of the best players in the world."
Macmillan does believes the Scottish champions should hold out for a fee in the region of the reported 6m for their prized -asset.
He said: "In this day and age, Bougherra is worth that. In today's climate, that's a reasonable asking price."
Teenage centre-half Wilson has been linked with Liverpool after impressing in his debut season for the senior team, meaning Rangers fans could be facing a double blow this summer.
Macmillan said: "If we lose both Bougherra and Wilson, it would be a major blow, albeit it would be softened a bit by Andy Webster coming back.
"He looks a much better player now (after his loan spell at Dundee United] and hopefully we will see the best of him next season. David Weir can't be expected to continue forever and both Bougherra and Wilson would be lovely assets to have on your books."