The German winger has attracted interest from a host of top European clubs after his impressive displays in the Bundesliga.
Havertz scored 20 times for Leverkusen last season, and Bild report United are now hopeful of persuading him to move to Old Trafford.
He recently suggested he was open to a move to England, telling Marca : “I think I’ve picked up enough international experience at Leverkusen.
“I know the rhythm of these international matches and I’m confident that I can go abroad at any moment and play at clubs outside of Germany.
“For the moment, I want to have a good season. Afterwards, we’ll see.”
Bayern Munich remain the likely frontrunners to land Havertz however, after he previously admitted a move to the Allianz appealed to him.
In a separate interview with the Spanish publication, he said: “They’re [Bayern] probably the biggest club in Germany and there are many players that were very successful at a young age that went there.
“Therefore, I get linked with Bayern, but we’ll see what happens. There are players that want to stay close to their family and others say, ‘I want to live in another country’.
“There’s something clear: if you want to be a star in Germany, you can’t go wrong with Bayern.”
United are believed to be keen on taking 20-year-old winger McNeil back to Old Trafford, having released the youngster just over five years ago.
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And Burnley boss Dyche said: “Look, we know Dwight is a player who will certainly go beyond us – but in a positive way.
“We don’t remotely want to lose good players. But eventually some players will outgrow the club in the sense of a bigger club coming along and buying them because of their skill set.
“Dwight is clearly a top talent. He is playing and delivering every week.
“If he remains grounded and keeps the hard work ethic he’s got – and I’m sure he will do that – then his talent will continue to shine through. And we’re not naïve enough to think it’s going to go unnoticed.”