Finance ministers Dana Reizniece-Ozola of Latvia, Pierre Gramegna of Luxembourg, Mario Centeno of Portugal, and Peter Kazimir of Slovakia all submitted formal applications for the job on Thursday, officials told Reuters.
The decision is expected to be taken on Monday.
The current Eurogroup chairman, Dutchman Dijsselbloem will step down on Jan 13 after two terms since 2012. He is no longer a finance minister after Dutch national elections earlier this year.
European socialists, of which Dijsselbloem is one, have been calling for the job to remain with a socialist because the European centre-right already has many of the top EU jobs, such as European Commission president.
Centeno is a socialist and has the backing of Italy, whose own finance minister Pier-Carlo Padoan also wanted the job, but dropped out because it is not certain he will stay in government after Italian elections due by May 2018.
The Harvard-educated economist has led Portugal during a strong recovery from the country's 2011-14 debt crisis and bailout. The country is growing at its fastest pace in at least a decade and the budget deficit is set to fall to its lowest in many decades.
In May, former German finance minister Wolfgang Schaeuble dubbed Centeno the "Ronaldo of EU finance ministers" -- a reference to the Portuguese soccer star -- as Portugal was about to exit the EU's disciplinary procedure for running excessive deficits.
Some in the Eurogroup say, however, say a minister from a country that had to be bailed out because of past policy mistakes would have smaller chance of getting the job. Many also say that Centeno has not contributed much to Eurogroup discussions in the past.
Slovakia's Kazimir, also a socialist, has run a tight fiscal ship at home and has been hawkish on bailouts for Greece. Some officials say that among socialist candidates he would have the backing of Germany and the Netherlands.