The EU crime-fighting agency Europol should jointly lead the investigation into the murder of a Slovak journalist, senior members of the European parliament have said, amid concerns that the Slovakian government is not doing enough to solve the crime.

Robert Fico’s government is on the brink of collapse, following the murder last month of the investigative journalist Ján Kuciak and his fiancee Martina Kušnírováand the biggest street protests in the country’s 25-year history amid widespread public distrust. His junior coalition partner has called for early elections.

Claude Moraes, a British Labour MEP who was co-chair of a European parliament delegation to Bratislava last week, urged Slovak authorities to set up a joint investigation with Europol into the double murder.

“All the indications are that it is an organised mafia-style killing and we think not enough is being done,” he told the Guardian. The Slovak authorities “have to open up the investigation, because there isn’t enough confidence that the government of Fico is investigating this openly and objectively”.

Outside experts needed to play a leading role in the investigation, Moraes said, because of the “existential elements”, the murder of a journalist and pre-existing public concerns about corruption and conflicts of interest.

Europol, the EU agency that fights cross-border crime, has already sent experts to Slovakia to help with forensics, while the FBI, Czech and Italian police are also assisting their Slovak counterparts.

The MEPs want Europol to co-lead a joint team, rather than be relegated to a support role that does not allow its officers access to all files.

On a two-day visit last week, the MEP group met Fico, who told them Slovakia was undertaking the biggest criminal investigation in its history. MEPs also met his key ally, Robert Kaliňák, who quit as interior minister on Monday, amid accusations of a conflict of interest.

The group visited the village of Veľká Mača, where Kušnírová and Kuciak were shot to death in the house they were renovating. The couple had planned to marry in May.

In a European parliament report to be published later on Tuesday, MEPs report a widespread perception among Slovak journalists and civil society that corruption allegations are not followed up, while EU money is seen as “a means to reward people close to the ruling party”.

The report does not draw conclusions, but following the visit, Ingeborg Grässle, the chair of the parliament’s budgetary control committee, said she had “concerns over alleged misuse of the EU funds”.

The leader of the centre-right European People’s party, Manfred Weber, said his colleagues feedback from the visit was worrisome. “For us it seems that the leadership of the Slovak government is not doing everything to clarify the two murders,” he wrote on Twitter.

On Wednesday the European parliament will debate the safety of journalists in Slovakia and allegations of misuse of EU funds – a central theme of Kuciak’s last story, which alleged that Italian businessmen with ties to the mafia had settled in eastern Slovakia and were raking in EU farm subsidies.

He alleged links between Calabria’s notorious crime cartel, the ‘Ndrangheta, and people close to the top of Slovakia’s government. A former model, who became Fico’s chief state advisor, Maria Trošková, was named in Kuciak’s investigation, as was her former boss, the secretary of Slovakia’s security council, Viliam Jasan.

“Two people close to the man who came to Slovakia as a person accused of being involved with the mafia in Italy have daily access to the prime minister of Slovakia Robert Fico, who chose them personally,” Kuciak wrote in the unfinished article, which was published posthumously.

The pair resigned from their roles pending an investigation into the murders, rejecting attempts “by some politicians and the media to link our names to these repellent crimes” .

The MEPs do not draw conclusions about the killings in their report, but noted the high mistrust among journalists and civil society groups of the government and courts.