Uefa, the governing body of football in Europe, is under pressure to investigate claims of massive corruption during Ukraine's preparations for Euro 2012, amid allegations that as much as $4bn (£2.5bn) in state funds allocated for the tournament was stolen by officials.

Rebecca Harms, the leader of the Green faction in the European parliament, said Uefa had to investigate why Ukraine cancelled competitive tenders for all Euro 2012 projects in 2010. Instead, contracts for building stadiums, roads and other infrastructure projects were awarded to a handful of shadowy companies, including one based offshore in Belize.

Opposition politicians allege that the companies belong, directly or indirectly, to government officials. Ukraine's government denies this.

Harms, a German MEP who visited Ukraine's second city, Kharkiv, last week, told the Guardian: "I will confront Uefa with these questions. I will also raise them in parliament. In whose private pocket did the money go? Uefa must take responsibility."

Ukraine embarked on a programme of modernisation ahead of Euro 2012. It built or renovated four stadiums; upgraded airports in the four host cities, Kiev, Lviv, Donetsk and Kharkiv, and laid or repaired 1,000 miles (1,600km) of roads. It also purchased a fleet of Korean high-speed trains to transport fans between venues.

The overall cost of Euro 2012 is in dispute. Borys Kolesnikov, the deputy prime minister, who in charge of infrastructure and Euro 2012, says the government spent $5bn, including $800m on stadiums. But others say the real figure including related projects and state guarantees is $10bn – more than the $2.25bn originally envisaged when Uefa gave the tournament to Kiev and Warsaw in 2007.

Ostap Semerak, Ukraine's shadow sports minister, said the then government of Yulia Tymoshenko wanted private investors to bear most of the cost. But after Viktor Yanukovych won power in 2010, defeating Tymoshenko, and subsequently jailing her, his new government largely abandoned private investment. About 80% of the Euro 2012 budget came from state funds, he said, with expenditure rocketing "sky-high".

Ukraine's parliament then cancelled competitive tendering for all state procurement contracts related to Euro 2012. A national agency in effect run by Kolesnikov was given power to award no-bid contracts. Kolesnikov says the exceptional measure was necessary because Uefa was on the brink in April 2010 of taking the tournament away from Ukraine – with stadiums unbuilt, preparations woefully behind schedule, and the country in crisis.

Semerak, however, alleges contracts were given to firms with indirect or direct links to government officials, and other figures associated with Yanukovych's ruling Party of Regions. Or, he says, they were part of a "corrupt pact" involving them.

Writing in the Kyiv Post on Friday, Semerak alleged: "The scheme is very simple: the official and his chosen contractor agree to share the budget cash. They cook up an overblown budget, and the contractor pays off a part of it back to the official as soon as he receives a transfer – usually in cash.

"Sources in such companies have said that the scheme allowed officials to receive (or, to put it bluntly, steal) between 30% to 40% of the state funds allocated for the tournament's preparations," he said, adding: "We're talking about up to $4bn."

Yuri Gromnytsky, Kolesnikov's press secretary, dismissed Semerak's claims of large-scale corruption as "science fiction". He said the government would invite an external auditor after the Euro 2012 championship was over to examine the accounts.

"They [the opposition] claim there is corruption. They should prove it," Gromnytsky told the Guardian. "Ukraine's preparations were a big success. Everyone sees it. There is a fantastic atmosphere in Ukraine. The fans are happy."

Semerak claims the theft of state resources took place with Euro 2012's most prestigious projects, including Kiev's Olympic stadium, which will host England's quarter-final against Italy on Sunday. Also under scrutiny is the new, purpose-built stadium in Lviv, airport terminals in Donetsk and Lviv, and roads. The Euro 2012 stadiums in Donetsk and Kharkiv, by contrast, were privately financed.

The reconstruction and development of the Olympic stadium cost the Ukrainian taxpayer $585m. Mark Rachkevych, a reporter for the Kyiv Post, who has investigated Euro 2012 corruption, said the bill was bigger than for other similar-capacity European stadiums. He said: "The Warsaw stadium was $35m cheaper. But it has a huge business centre attached." Munich's Allianz Arena also cost less, despite the fact that Ukraine has cheaper labour costs than Germany.

Rachkevych added: "Euro 2012 was an amazing opportunity [for Ukraine] to really scale up the infrastructure projects by including public-private partnerships and private investors. But they [the government] decided to do it on their own, mostly with taxpayers' money, in a non-transparent way. The question remains: how much money was really spent, and how much went into people's pockets?"

Last week the Olympic stadium's general contractor, Volodymyr Artiukh, confessed to helping embezzle $3m of public money from a state bank. Artiukh was already under investigation for alleged fraud when in June 2010 Kolesnikov picked his company, AK Engineering, to reconstruct the stadium.

The firm also revamped the nearby sport's palace, used as the Euro 2012 accreditation centre. According to documents obtained by the Ukrainskaya Pravda newspaper, Artiukh has long-standing ties to Kolesnikov. Artiukh was in partnership with the minister's lawyer, Ivan Shakurov, the paper alleges, though Artiukh himself denies any connection to the minister.

Kolesnikov denies any connection to AK Engineering or Artiukh. "It's absolutely not true," Gromnytsky said. The deputy prime minister has conceded that some Euro 2012 projects cost more, arguing that given the tight deadlines involved there was no alternative. "The greedy person pays twice," he said last year, citing a Russian proverb.

Semerak, however, alleges that Kolesnikov removed companies already working on Euro 2012 construction sites, replacing them with other companies connected to him "in one way or another".

There are also unanswered questions about a second mysterious firm, Altkom. Altkom is a large financial-industrial group based in Donetsk, the home region and political heartland in eastern Ukraine of President Yanukovych. Altkom received $800m from the Euro 2012 state budget. It was given no-bid contracts to build an airport runway in Donetsk ($225m), the Lviv stadium ($175m) and several roads.

It is unclear who actually owns Altkom. Formally, it belongs to Eurobalt Limited, a company registered in Newhall Street, Birmingham. Eurobalt is in turn owned by a mysterious offshore company based in Belize, Trinitron Investments.

Last year the Organised Crime and Corruption Reporting Project tracked down the company's nominal "director" and discovered that she was a yoga teacher based in Cyprus.

The teacher, Lana Zamba, was the firm's proxy director until last December. (She was removed soon after her identity was made public) She was also a "director" of 23 other British firms. In reality she has nothing to do with any of them, her husband told the Kyiv Post, and received a token salary of just $534 a month.

Zamba was born in Ukraine, received a Cypriot passport in 2006, and also "fronts" for several companies owned by Russians.

Asked who might be Altkom's ultimate beneficiary, Gromnytksy said: "I don't know who. Nobody seems to know who owns Altkom. But there is no direct connection between someone from the Ukrainian government and Altkom. The firm did a lot of construction. There was a transparent procedure. There is no dark story here."

Asked if Altkom was indirectly connected to government, he replied: "We don't have any information on that."

Semerak told the Guardian he was writing this week to the home secretary, Theresa May, asking her to investigate Eurobalt Ltdon suspicion of alleged money-laundering. Last month Tymoshenko and Semerak contacted the Financial Action Task Force (FATF) and asked it to examine alleged Euro 2012 corruption in Ukraine. Semarak said he approached the inter-governmental body because Ukraine's prosecutor was not independent.

In a recent paper Poland's Centre for Eastern Studies asserted that it was unclear "who really derives the profits" from Altkom's activities. It concluded: "The Ukrainian media most often mention Kolesnikov.

"However, given the scale of orders carried out by Altkom, it seems more likely that it is connected to a larger group of Ukraine's most senior state officials and businessmen linked to [Yanukovych's] Party of Regions." It noted that previous Ukrainian governments had handed lucrative state contracts to trusted "friends". Altkom has denied links to any government officials.

Government insiders privately admit that money did disappear in the preparations for Euro 2012. But they stress that stadiums were all built on time to Uefa specifications – an epic feat against an almost impossible timetable. "We have a result," one source stressed. The insiders also say that the corruption was on a far smaller scale than in neighbouring Russia, where as much as 100% of the state budget can allegedly be stolen, especially with military procurement contracts.

Serhiy Shcherbina, an investigative reporter with Ukrainskaya Pravda, said his newspaper's allegations of Euro 2012 corruption were backed by evidence and research done over many months. He said: "There are stadiums, airports. Things were built. It's a fact. You can't dispute it. But the question remains: how were they built?"