Sprint agreed to pay Blackboard millions of dollars last year to offer a free mobile version of its popular course-management software that would work over Sprint’s network, but not those of competitors like Verizon or AT&T.

Sprint officials reasoned that colleges would flock to the free version of the software, Mobile Learn. And because only Sprint customers could use the software with Android, BlackBerry and Palm phones, college students would have a powerful incentive to sign up with Sprint.

But the deal apparently didn’t work out as Sprint had hoped. The communications giant sued Blackboard last month, alleging that Blackboard’s iPhone and iPad versions of Mobile Learn violated Sprint’s right to exclusivity because they can access the Internet over campus wireless networks. Sprint has also withheld nearly $3-million in referral fees from Blackboard since May, according to legal filings.

Blackboard officials say they believe the contract precludes their application only from using the cellular signals of Sprint competitors. Allowing students to use campus Wi-Fi networks is allowed, the company argued.

The lawsuit was first reported on Monday by Seeking Alpha, a stock-market-analysis Web site.

If Sprint’s suit is successful, it could threaten one of Blackboard’s most lucrative partnerships and cause major headaches for colleges that have come to rely on the free version of Mobile Learn. Sprint asked a judge to bar Blackboard from allowing its free iPhone and iPad versions to use wireless networks.

Even if the lawsuit fails, the strained relationship between the two companies could foretell major changes for Blackboard’s popular free Mobile Learning application. Sprint has essentially subsidized the application by paying Blackboard millions of dollars in referral fees.

Matthew Small, Blackboard’s chief business officer, said in an e-mail that the company would not have any comment, “except to note that as the matter is reviewed, it will have no impact on our clients and their full and uninterrupted use of our mobile applications and services, including Mobile Learn for Sprint.”

Blackboard was less circumspect in court, calling Sprint’s lawsuit “a transparent attempt to avoid its contractual obligations to Blackboard.”

“The partnership has turned out to be less profitable for Sprint than Sprint hoped,” Blackboard said in an August 1 response to the lawsuit. Mobile Learn for Sprint “has proven an insufficient incentive to persuade many college students to sign up with Sprint. … Instead, students continue to switch away from Sprint to Verizon and AT&T, and Sprint’s market share continues to decline.”

Sprint did not respond to a request for comment late Monday. In its lawsuit, Sprint said that it secured the right to be the “sole network” on which a free version of Mobile Learn would run. A wireless network qualifies as a network in addition to a cellular network, Sprint says, meaning Blackboard’s free wireless-enabled iPhone and iPad applications should not be allowed.

Blackboard responded that Sprint not only agreed to the wireless functionality but consistently advertised it to its customers. “After 18 months of approving, endorsing, and promoting the very conduct of which it now complains, Sprint asks the court to enjoin Blackboard from making Blackboard Mobile Learn available over Wi-Fi at no cost to schools,” Blackboard said.

As Sprint has apparently struggled to make the deal work, it has paid Blackboard handsomely for the privilege. The exact amount is unclear, but Sprint will eventually owe more than $50-million that it has not already paid, Blackboard said in a court document. Sprint must pay Blackboard $2 per month for every student who has downloaded Mobile Learn for Sprint, in addition to millions in other fees, the document says.