The move comes ahead of a government decision on whether to change the laws around tipping following an investigation by the Department for Business, Innovation and Skills.
GMB said that Turtle Bay - which has 19 UK restaurants - requires staff to pay back three per cent any sales they generate to the restaurant manager at the end of their shift.
This means that a waiter who sold £1,000 of meals and drinks in a shift would have to pay £30 in cash back to the restaurant.
The Guardian reported last year that staff at Turtle Bay were still required to pay the charge even when they had not made enough in tips to cover it.
Now GMB is meeting with Stella Creasy, Labour MP for Walthamstow and Nick Bowles MP, Minister for Business, Innovation and Skills on 26 January to discuss the issue.
Perry Phillips, GMB organiser, said: “GMB will make clear that we want Turtle Bay to stop charging their workers three per cent of the table sales they accumulate each shift that they have to give back to the restaurant manager at the end of their shift.
“There should be a clear policy that all tips should go to the workers. There is no justification for these monies going to cover costs of running restaurants. These costs have to be recovered by upfront prices charged to customers."
The union is urging Turtle Bay to follow Las Iguanas and Pizza Express in changing their policy to ensure all tips go direct to staff.
However Turtle Bay claimed that it was ‘not aware of any occasion’ where a waiter had not earned any tips.
The company said in a statement: “The three per cent is a calculation on a server’s total sales, and in the vast majority of cases customers will leave in excess of ten-15 per cent of their bill as tips. In addition, a ten per cent service charge is added to all tables of five or more, which the server also keeps.”
Last year restaurant groups including Casual Dining Group - which owns Bella Italia, Cafe Rouge, and La Tasca - bowed to public pressure to change their tipping policies after it was revealed they were taking a percentage of staff tips as an administration charge.