In an interview broadcast on Sunday evening, Mr Trump admitted that Vladimir Putin was “probably” behind such assassination attempts.
But he said that he did not need to be any tougher on the Russian leader because “I rely on them ... it’s not in our country”.
Speaking to the CBS 60 Minutes programme, Mr Trump denied that he was afraid to criticise the Russian regime.
“People don’t understand why you never have a harsh word for Vladimir Putin,” Lesley Stahl, the veteran American journalist asked him during the programme. “I don’t understand it.”
Mr Trump rejected this by arguing that he had sent Ukraine weapons and “tank killers” to aid its war against Russian-backed separatists in the east. “Obama didn’t. You know what he sent? He sent pillows and blankets.”
In 2014, Mr Obama did turn down Ukrainian requests for weapons, but instead agreed a $53m package of aid, including $46m of non-lethal military equipment such as body armour and night-vision goggles
But when pressed on whether he believed Mr Putin was personally involved in directing Russian assassination attempts and poisonings, Mr Trump agreed.
“Probably he is, yeah. Probably. But I rely on them, it’s not in our country.”
The president also appeared to deflect criticism of his relationship with Mr Putin by suggesting that China also interfered in the 2016 election.
When asked if he believed Russia was involved with his election victory, he said: “They meddled. But I think China meddled too. And I think, frankly, China is a bigger problem.”
While America’s intelligence agencies are unanimous in believing that the Putin regime tried to influence the election in favour of Mr Trump, no evidence of any Chinese interference has emerged.
Ms Stahl accused Mr Trump of trying to “divert” her questions about his closeness to Russia, but he denied this. “I’m not doing anything. I’m saying Russia, but I’m also saying China.”
Despite Mr Trump’s ambivalence over the Salisbury novichok attack, his government has taken steps with the UK against Russia.
In August the State Department announced it would impose fresh sanctions because it agreed with Britain’s assessment that Moscow had used a chemical weapon in the attack on their former spy Sergei Skripal and his daughter Yulia.
A second batch of stiffer sanctions would be launched after 90 days if Russia did not allow United Nations inspections and promise not to use chemical weapons again.
At the time, a spokesman for Downing Street welcomed the “strong” response to the Salisbury attack by “our US allies”, which would “send an unequivocal message to Russia that its provocative, reckless behaviour will not go unchallenged”.