Mr Trump tweeted that it would be premature to consider any relaxation "until the Ukrainian and Syrian problems are solved".
The president also said he would work "constructively" with Russia after meeting President Putin last week.
Russia meanwhile has insisted that it will not change its policies in Ukraine and Syria because of the sanctions.
"Our policy on Syria and on Ukraine has never been and will never be determined by the pressure of sanctions applied by the USA," parliamentary international affairs spokesman Konstantin Kosachev said.
He said that US policy towards Russia was not determined by the requirements of international relations but by "the zigzags of US domestic policy and [the] confrontation between [President] Trump and Congress".
US Secretary of State Rex Tillerson, who is visiting Ukraine, on Sunday called on Russia to take the "first step" to bring an end to the separatist conflict in the east of the country.
He said the sanctions against Russia would remain "until Moscow reverses their actions".
The secretary of state said it was incumbent on Russia to make sure the Minsk peace agreement, which  for Ukraine was fully implemented.
The war in Ukraine, along with Russia's annexation of Crimea in 2014 and its military backing of Syrian leader Bashar al-Assad, have strained ties between the two countries.
US Senators last month voted overwhelmingly to impose fresh sanctions on Russia over its alleged interference in last year's US election.
The senators also agreed to set up a process by which Congress could block any attempt by President Trump to scale back the sanctions.
US sanctions imposed against Russia over its annexation of Crimea are to remain, President Donald Trump says.