The 29-year-old made his first competitive outing in almost two-years in last Sunday's Carabao Cup final.
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Karius could not prevent both Casemiro and Marcus Rashford from scoring, as Manchester United secured a 2-1 victory and ended their six-year trophy draught.
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The goalkeeper joined Eddie Howe's side as third-choice shot stopper behind Nick Pope and Martin Dubravka, but featured given both were ineligible.
According to FCInternews, Luciano Spalletti's side could offer the German the opportunity to ply his trade between the sticks in Italy.
The reports add Inter are on the lookout for a new goalkeeper as they seek a replacement for Samir Handanovic, who is expected to retire at the end of the season after 11 years of service.
Karius, who is out of contract at Newcastle in the summer, could be persuaded by the move, given his girlfriend, Diletta Leotta, lives there.
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She works for DAZN and covers Serie A as a television presenter, and both were enjoying time in Milan last week ahead of the final.
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Diletta was in attendance at Wembley to watch Karius in action against the Red Devils before returning home.
"I just tried to do my job and help the team. I think I played pretty well for the time I was out for," Karius said after the defeat to Man United.
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"It's not easy as a goalkeeper to get straight in the rhythm in a game like this. So, I think I did quite well but ultimately, of course, I wish we would have brought home the cup.
"It's the worst feeling to go up there [to collect your medal] and be applauded when you're not winning. We came into this with a lot of confidence and a lot of belief that we could do it.
"We thought we were the team that can bring them down but in the end, it wasn't meant to be.
"The luck in small moments wasn't on our side and you need a bit of luck to win a game like this when it's really 50-50, so everyone's really disappointed but we have a lot of quality players, a lot of young players that will play many more games like this in the future."