The Sunday Mirror claims PSG are willing to let Draxler leave for a transfer fee of around £32m ($41m), as they pursue the signing of teenage striker Kylian Mbappe from Monaco.
The Ligue 1 club have already strengthened their attack with the world-record signing of Neymar from Barcelona and are reportedly open to selling Draxler as he is surplus to requirements.
Draxler did not feature in PSG's 3-0 win over Guingamp last weekend due to injury. He joined the French side from Wolfsburg in a £40m move in January.
The Germany international's agent has denied that PSG are looking to sell Draxler, saying in an interview last week that he had received no information from the club to that effect.
"We are not interested in whether [French daily] L'Equipe wrote something," Roger Wittmann was quoted as saying by ESPN. "There has been no change, it is not a topic.
"If somebody writes, 'I have this information from [PSG chairman] Nasser Al-Khelaifi,' then I will consider it and call the president to speak about Julian's situation.
"Once again, though, this is not the case. So, everything has been said."
A source told ESPN that PSG had never considered selling Draxler and that head coach Unai Emery was happy with how the attacking midfielder had started his career in the French capital.
Liverpool could pursue Draxler if they end up selling Philippe Coutinho to Barcelona, while United could target the Germany star as an alternative to Inter Milan winger Ivan Perisic.
The Reds turned down a £114m bid from Barcelona for Coutinho last week and have remained adamant that the Brazil international will not be sold.
Coutinho scored 13 goals in 31 league appearances for Liverpool last season to help the club qualify for the Champions League.