Theresa May should tell Donald Trump during his visit to Britain this week that he will wreck his own mission to “make America great again”, and instead empower the Kremlin, if he continues to undermine Nato, the former Tory foreign secretary, Sir Malcolm Rifkind, has said.

Senior UK politicians and diplomats believe the next 10 days of international diplomacy – during which the US president will also attend a Nato summit in Brussels and hold talks with President Vladimir Putin in Helsinki – will be vital in determining the future strength of the western alliance as a bulwark against Russian aggression.

Trump arrives in Britain on Thursday for a three-day visit which will be met by protests across the country, including a march in central London on Friday. At the Nato summit, he is expected to urge members of the alliance to increase their defence spending.

Rifkind told the Observer that May should mix flattery about his position as leader of the western world’s leading economic and military power with a stern warning when the two leaders have a bilateral meeting, probably at Chequers on Friday. “He is currently the leader of the most powerful country in the most powerful military alliance there is. That is Nato,” said Rifkind.

“She should impress on him that he mustn’t kill the goose that lays the golden egg. If he wants to punish Nato countries for not paying enough to the alliance and that leads to its disintegration, we will all suffer but for him as well it will be utterly self-defeating. Who will take the greatest pleasure and gain most from the disintegration of Nato? It will be the gentleman in the Kremlin.”

Rifkind said senior Republican politicians whom he met on a visit to the US last week told him that if Trump continued to criticise Nato and weaken US backing for it it would also seriously damage support for him within his own party.

Trump’s lukewarm support for Nato has led some European allies to question if the US would come to their aid if Russia were to attack them. Before the Nato summit on Wednesday and Thursday many Nato and EU member countries believe any sign of Nato disunity will be exploited by an increasingly aggressive Moscow.

Sir Nigel Sheinwald, former UK ambassador to Washington, said the leverage May would have in talks with Trump would depend on his approach at the Nato summit in Brussels before he flies to the UK. “A lot will depend on how Trump emerges from the Nato summit. She can’t reverse whatever impression he has left there. If he wants to come over as a qualified supporter of Nato, she has something to play with in her bilaterals with him. If his time in Brussels is spent confirming the idea that he wants to undermine Nato then she has a much more difficult time on her hands.”

If Putin detects any splits in Nato when he meets Trump, he will be sure to exploit them Labour MP Chris Bryant

Sheinwald said May would want to stress that Nato “depends on American leadership and on Putin not being able to divide either America from Europe or the European countries among themselves”.

However, he questioned whether Trump really believed that his power on the world stage rested in the US’s traditional alliances. “The fundamental question is whether Trump buys into that basic idea that making America great depends on America using its fundamental assets in the world, which are its alliances. So far, it has not been clear that he does buy into that. The Nato summit will be crucial in that regard.”

Labour MP Chris Bryant, a member of the foreign affairs select committee, said May must impress on Trump the need to stress Nato unity in his meeting with Putin. “Any slither of difference between us and the US on this is extremely dangerous. With the Russians increasingly deploying submarines in the north Atlantic, Moscow poses a serious threat. If Putin detects any splits in Nato when he meets Trump, he will be sure to exploit them.”