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.
But the Red Devils face a fight for his signature, with Real Madrid, Barcelona and Liverpool all rumoured to be interested in the 20-year-old.
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.”