The eight-time world champion has been enveloped in gossip that he will switch to the Gresini Ducati squad in 2024 as Honda's problems in MotoGP persist, though Marquez has continually stated "I have a contract" with HRC.
Monday's post-race test is seen as vital in Marquez's decision to remain or leave Honda, with the Japanese marque bringing a prototype 2024 bike, which is already being ridden by wildcard Stefan Bradl at the San Marino Grand Prix.
Marquez said on Friday that he felt the team was "reacting" to his comments, noting how vital engineers are to a test team.
The Tokyo factory has launched a campaign to hire technicians from other teams, several of them European, to try to turn around a calamitous situation that could get much worse if Marquez decides to leave.
That is what Puig, the man in charge of this recruitment mission, is trying to avoid.
The Honda formula does not serve to correct the lack of traction and acceleration of an RC213V that, in the hands of Marquez, is only capable of recovering ground under braking, with the risk of crashing.
Marquez, who is coming out of the most complicated period of his life after struggling with injury, is tired of believing in the impossible, and is willing to explore a change of scenery.
On Monday he is scheduled to ride the prototype of the bike planned for 2024. Until then, Puig repeats what he has said so far with some nuances.
"We have a contract with Marc, and he hasn't told us anything," he told DAZN.
"It is obvious that our situation is bad. It is very important to give him a bike he can trust next year.
"It is Honda who has to convince Marc to stay. And not in words, but with facts."
Puig empathises with Marquez and understands perfectly that he values any scenario.
"I perfectly understand the frustration Marc may be going through, and also that he talks to other brands," he said. "At a technical level, it is clear that we are quite far from finding the solution."
Puig understands that such a dramatic situation as the one currently affecting HRC can only be addressed with a powerful shake-up.
He added: "When you change something, you can change it little, which is like doing nothing, or you can make a radical change.
"That's why we are thinking about a radical change," said Puig, who again made it clear that if Marc decides to leave, no one will block him. "Honda is not a company that forces anyone to race for it," he added.
