The demand was made at a news conference in Parliament on Tuesday by Evelyn Regner and Othmar Karas, both Austrian MEPs.
They have drafted a report that seeks more transparency in the way multinationals conduct their businesses.
The report says such companies should be compelled to publish information on where they make profits and how much tax they pay in the EU, on a country-by-country basis.
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The draft will be voted on by the economic and legal affairs committee on Tuesday before going to the plenary in Strasbourg next week.
Regner, a Socialist member, said the estimated cost of lost tax revenues in the EU is €50bn to €70bn per year.
She said that over the past 20 years, corporate tax levels had fallen from 35 per cent to 22 per cent and this had created a shortfall in national tax revenues.
But the burden of meeting this shortfall, she said, had fallen on SMEs, while large multinationals were "getting away with not paying their fair share" of tax.
Regner believes that a country-by-country system of reporting tax affairs would close loopholes in existing rules which allow large companies to do business in Europe but "shift their profits out of the EU."
She said, "We want to reveal the dark side of how these companies operate and this draft law will do that. It will give us precise information on where a company is based, what they are doing, what tax they pay and where they pay it."
Her comments were echoed by Karas, an EPP group member, who said that country-by-country reporting would lead to more transparency and openness.
Karas said the proposed new transparency rules would mostly apply to only those companies with an annual turnover of €750m or more, or about 6000 companies in Europe.
He pointed out that similar efforts to improve transparency in the banking sector were introduced in 2015 and the aim now was to apply the same conditions to multinationals.
"We need to close the loopholes and gaps in tax laws that allow big companies to pay so little tax," he said.
He added, "Much of the public face a heavy tax burden and SMEs pay up to 25 per cent in corporate tax but multinationals are escaping with paying, in some cases, less than one per cent in tax.
"Not enough pressure is being brought to bear on big companies but parliament now has co-decision powers in this area and intends to use those powers."
Karas said, "Companies need to be crystal clear about their turnover and where their income is being generated."
Both MEPs are members of Parliament's committee of inquiry to investigate alleged contraventions and maladministration in the application of Union law in relation to money laundering, tax avoidance and tax evasion.
They said the report on country-by-country tax information had faced some resistance in certain quarters, including the EPP group.
But Regner said, "It was felt that the rules would be bureaucratic but this is not true and I think we have managed to allay such concerns."
She said that if, as expected, Parliament backs the proposals in full plenary, the matter will then move to a trialogue with the Council.
"At present, member states appear to be poorly motivated to do something in this area but we obviously hope that will change," she added.