The pair first clashed Friday at an EU meeting in Vienna when Salvini referred to African migrants as “slaves”. His remarks prompted an angry outburst from Asselborn who has defended immigration as necessary to counter Europe's ageing population.
Salvini later shared a video of the row on his Facebook page along with comments aimed at further taunting the Luxembourgish minister.
In an interview with German media, Asselborn denounced Salvini for using “the methods and tone of the fascists from the 30s”.
“I stand by what I said,” he told Spiegel Online this weekend, adding: “It was a calculated provocation.”
Salvini, whose anti-immigration League party is part of Italy's ruling coalition, hit back on Sunday.
“The Socialist minister of the fiscal paradise of Luxembourg calls me a 'fascist' today after comparing our Italian emigrant grandparents to today's illegal migrants and after interrupting my speech,” the 32-year-old tweeted.
“If he likes immigrants so much, he can have them all, we've already received too many in Italy.”
Despite a sharp fall in the numbers of asylum-seekers in Europe since the crisis erupted in 2015, the issue remains one of the most contentious within the EU and is expected to be high on the agenda at an informal EU summit in Austria this week.
Frontline state Italy has adopted a much tougher anti-migrant stance since the League formed a government with the anti-establishment Five Star Movement in June.
“In Italy we feel it's necessary to help our children make more children. And not to have new slaves to replace the children we're no longer having,” Salvini was filmed saying at Friday's EU meeting, which was closed to the press.
This is a remarkable video showing the interior ministers of Luxembourg having a go at Italy's Salvini over his uncompromising policy migration. Lays bare EU divisions. “Merde alors!” the Luxembourg min says at one point. https://t.co/S2zRHGuZWt
— Crispian Balmer (@crispiandjb) September 14, 2018
Asselborn, seated two places down from Salvini, can be seen exclaiming in the video: “That's going too far!”
Unfazed, the Italian interior minister continued: “If you in Luxembourg need more immigration, I prefer to keep Italy for Italians and that we start having children again.”
Asselborn, visibly agitated, interrupted Salvini at this point.
“In Luxembourg, sir, we have dozens of thousands of Italians! They came as migrants, they worked in Luxembourg so you in Italy would have money for your children,” he said, adding: “Merde alors [goddammit]!”.