John McGuinness, chairman of the Oireachtas finance committee, invited the chief executive of the tech giant to answer questions on the European Commission ruling next month.
Sinn Féin TD Pearse Doherty said he had been told the offer had been rejected and branded Mr Cook’s rejection a snub to the Irish people and parliament.
“Tim Cook has a moral obligation to come before the committee,” he said.
“I’d urge Government ministers to encourage Tim Cook to respond favourably to the request. It’s a snub to the Irish people and a snub to the Irish parliament.”
Mr Doherty said Apple had told the committee that Mr Cook would not attend because the company is appealing the tax ruling through the European courts.
The EU competition commissioner Margrethe Vestager, who issued the ruling on Apple tax bills last year, is due to attend the committee, despite also being a party to the appeal case.
The watchdog’s landmark ruling into the iPad and iPhone maker found it paid just €50 in tax for every €1m of profit made outside the US in 2014 after getting assurances from tax inspectors about a complex corporate structure in 1991 and 2007 and how it routed sales figures through subsidiaries in Ireland and on to the US.
Mr Cook previously appeared at a congressional hearing in Washington DC where he explained that Apple effectively paid a 2% tax rate for its subsidiaries in Ireland, where the headline corporation tax rate is 12.5%.
Apple has about 6,000 employees in Ireland, many based in Hollyhill, Cork, and is planning to build a huge data centre near Athenry, Co Galway.
The company wrote to the committee on December 20 last year and rejected the invite, stating that the issue was “complex and challenging”.
Apple is understood to have said that contacts were ongoing with the Department of Finance, the Revenue Commissioners, and the European Commission, and that given “the sensitive nature” of the issue it had been advised not to do anything that could prejudice it.
Mr McGuinness described Mr Cook’s rejection as “more than disappointing”.
“Apple is a huge employer here. It’s not just a question of Tim Cook being given an invitation it’s about the profile of his company and its attitude to the Irish Government,” he said.
“I would expect a lot more from his company.”