The Belfast advertising agency AV Browne is bringing the case after missing a deadline in a bid for work.
The contract is for advertising and design work for a period of two to four years.
It should have been awarded earlier this year, but the legal action has stalled the selection process.
The court was told AV Browne's problem arose with the Department of Finance-run procurement website on Wednesday 14th October 2015.
The agency's lawyer said just before a 3pm deadline "the portal froze" and "a figure could not be entered into a box".
It resulted in AV Browne's tender not being accepted.
The agency has worked for Tourism NI for several years, including on its major 2012 campaign, 'Our Time, Our Place'.
Three other companies are vying to take over, but AV Browne is fighting in court to remain in contention.
Belfast High Court heard the agency had complied with the initial tender deadline of September 18 and that the issue arose with additional information being provided at a second stage.
An AV Browne employee was using an Apple computer to submit the information.
The agency's lawyer said: "The instructions on the tender website only dealt with Windows users on PCs.
"Apple computers were disadvantaged.
"When the box on the website was clicked it just froze at zero."
He argued that regardless, AV Browne had complied to the only deadline stated in the tender documentation, which was 18 September, and that discretion could be exercised.
The case is due to last two days.
A £14m advertising contract with Tourism NI is the subject of court action after Belfast High Court was told a government website froze.