Donald Trump told Vladimir Putin he had to act tough next to the Russian president for the cameras, according to the former White House press secretary Stephanie Grisham.

“OK, I’m going to act a little tougher with you for a few minutes,” Grisham says she heard Trump tell his Russian counterpart in Osaka in 2019. “But it’s for the cameras, and after they leave, we’ll talk. You understand.”

Grisham makes the claim in a new book, I’ll Take Your Questions Now, which will be published next week. The Washington Post obtained a copy.

Trump’s presidency was dogged by his relationship with Putin, the focus of the special counsel Robert Mueller’s investigation of Russian election interference and links between Trump and Moscow.

Mueller did not establish a conspiracy but stressed that he did not exonerate Trump of seeking to obstruct justice. Speculation over the two leaders’ relationship remained rampant, particularly over a meeting alone save for interpreters in Helsinki in 2018.

In front of the media at the G20 summit in Osaka in 2019, with Grisham sitting nearby, Trump joked with Putin that they should both “get rid” of journalists who published “fake news”, saying: “You don’t have this problem in Russia.”

Putin said: “Yes, yes, we have too, the same.”

Trump later smirked, pointed at Putin and said: “Don’t meddle in the election.”

Grisham was Trump’s third press secretary, an unhappy reign in which she did not hold a single White House briefing. Her book has been extensively trailed, titbits including a comparison of Melania Trump to Marie Antoinette.

A spokeswoman for Donald Trump has called the book “another pitiful attempt to cash in on the president’s strength and sell lies about the Trump family”.

Melania Trump has also dismissed the book. Grisham was working for the first lady when she resigned on 6 January this year, the day of the deadly attack on the US Capitol by Trump supporters.

According to the Post, Grisham says Donald Trump “was too wrapped up in his own ego and his own delusions about his invincibility” to deal properly with the coronavirus pandemic.

Discussing notorious events including “Sharpiegate”, involving a crudely doctored hurricane map, and the porn star Stormy Daniels’ claim that Trump’s penis was small and shaped like a mushroom, Grisham writes about White House incompetence extending to how staff lied to Trump to stop him losing his temper.

One such instance, Grisham says, came in 2018, when Air Force One was customarily made available to carry the casket and service dog of former president George HW Bush, the deceased patriarch of a family Trump loathed.

“We knew he wouldn’t be OK with that, even for a brief trip,” Grisham writes. “Dead bodies, death, sickness – those things really seemed to creep him out.”

Grisham is heavily critical of Melania Trump, who she says began to break with her husband over the Stormy Daniels affair.

Writing about how Melania was criticised for not standing with her husband on election night, as Jill Biden did with Joe Biden, Grisham says Melania said: “I don’t stand next to him because I don’t need to hold him up like she does. Can you imagine?”

“That made me laugh,” Grisham admits.