The bombardments by Russia are said to have been some of the largest and deadliest attacks since the start of the war, now in its fourth year.
Russian strikes in Kyiv, Ukraine's capital, killed at least 13 people and injured dozens more, including children, over the weekend.
And by Wednesday, the attacks had shown no signs of slowing down.
In Trump's remarks about the escalation of violence and whether he thinks Putin is serious about ending the war, Trump said: "I'll let you know in about two weeks."
"Within two weeks. We're gonna find out whether or not (Putin is) tapping us along or not.
"And if he is, we'll respond a little bit differently."
The comments are a sign of Trump's growing frustration, as the White House's repeated efforts to negotiate a deal between Russia and Ukraine appear ever more futile.
This includes a recent two-hour phone call between Trump and Putin, after which the US president said the discussions went "very well".
Putin walked away from the call saying he was ready to work with Ukraine on a "memorandum on a possible future peace agreement".
That call was one week before Russia launched hundreds of drones and dozens of missiles towards Ukraine's capital, according to Ukraine's air force.
And a memorandum has yet to be produced by Russia.
So far, Trump's threats have not appeared to concern Moscow sufficiently for it to concede to his demands. Trump has not delivered on previous such threats.
Since taking office, Trump has only taken action against Ukraine, as Washington sought to steer the countries to Trump's demand for a truce.
This included an eight-day suspension of US military assistance and intelligence sharing with Kyiv in March.
Meanwhile the US administration has not publicly demanded any significant concessions from Russia.
The White House rejects accusations of appeasing Moscow or failing to enforce its will, pointing out that all the Biden-era sanctions remain in force against Russia.
But so far its mediation approach appears to have made the Kremlin more, not less, empowered.
After the latest attacks, Trump wrote on Truth Social that "something has happened" to Putin, which the Kremlin said were comments made "connected to an emotional overload".
Russia's attacks on Ukraine continued in the days afterwards. Trump then escalated his criticism. On Tuesday, he said Putin was "playing with fire" and that "lots of bad things" would have happened to Russia if it were not for Trump's involvement.
A Kremlin aid responded to the latest Trump Truth Social post by saying: "We have come to the conclusion that Trump is not sufficiently informed about what is really happening."
Putin aide Yury Ushakov told Russian state TV channel Russia-1 that Trump must be unaware of "the increasingly frequent massive terrorist attacks Ukraine is carrying out against peaceful Russian cities."
On Wednesday, Germany's new chancellor, Friedrich Merz, told Ukraine's Volodymyr Zelensky that Berlin will help Kyiv produce long-range missiles to defend itself from Russian attack.
The Kremlin has warned that any decision to end range restrictions on the missiles that Ukraine can use would be a dangerous change in policy that would harm efforts to reach a political deal.
In the face of Russia's recalcitrance, Trump has frequently softened his demands, shifting the emphasis from his original call for an immediate 30-day ceasefire, to which only Ukraine agreed, to more recently demanding a summit with Putin to get what he says would be a breakthrough.
Putin and his foreign minister Sergei Lavrov have upped their demands from earlier positions since the US restored contacts with the Russians in February.
These have included a demand that Ukraine cede parts of its own country not even occupied by Russia and that the US recognises Crimea as a formal part of Russia.
Michael McFaul, a former US ambassador to Moscow, calls this a "poison pill" introduced by Russia: Creating conditions Kyiv could never agree to in order to shift blame onto Ukraine in Trump's eyes.
The war has claimed tens of thousands of lives and left much of Ukraine's east and south in ruins. Moscow controls roughly one-fifth of the country's territory, including Crimea, which it annexed in 2014.
Zelensky has accused Moscow of delaying the peace process and said they were yet to deliver a promised memorandum of peace terms following talks in Istanbul. Peskov insisted the document was in its "final stages."