“From the perspective of the United States, the time is now to finalise Sweden's accession,” Mr Blinken said alongside Swedish Prime Minister Ulf Kristersson in northern Sweden.
“We will continue to work to complete Sweden's accession by the time our leaders gather in Vilnius for the Nato summit.”
Any new Nato member must be unanimously approved by the 31 alliance countries, but Sweden’s bid to join has been held up by Turkey and Hungary.
Ankara claims Stockholm is harbouring Kurdish militants, while Budapest's far-right government has been angered by Swedish criticism over what it claims are moves away from democracy.
Sweden denies harbouring members of Kurdish groups that Turkey considers to be terrorists and has taken steps to appease Ankara, including by passing an antiterrorism bill aimed at addressing some of its concerns.
“We acknowledge the fact that they have had good reasons to have had concerns,” said Mr Kristersson.
He said the legislation would go into effect on June 1.
Russia’s invasion of Ukraine forced Sweden and Finland to rethink their long-standing and deeply rooted neutrality, ultimately leading both to apply for Nato membership.
Finland was approved in March, but Sweden has still not received the go ahead.
“Sweden again and again has proven its commitment to Nato, to its values, to its mission to its members,” Mr Blinken said.
“[It is] a strong, vibrant democracy with highly capable forces that have been working shoulder to shoulder with Nato members for decades.”
On Monday, US President Joe Biden suggested that Turkey’s acceptance of Sweden may hinge on acquiring F-16 fighter Jets.
He said he spoke to Turkish President Recep Tayyip Erdogan to congratulate him on his victory in Sunday's run-off election.
“He still wants to work on something on the F-16s,” Mr Biden said. “I told him we wanted a deal with Sweden, so let’s get that done.”
Turkey is seeking F-16s after it was dropped from the F-35 next-generation fighter programme because it bought Russian air defence systems.
The White House sought to walk Mr Biden's comments back on Tuesday, clarifying that the potential sale of F-16s to Turkey was not conditional on it approving Sweden's membership.
“When it comes to any conditions as it relates to the sale of F-16s to Turkey, and with them approving Sweden's bid to join Nato, that is not a condition,” White House spokeswoman Karine Jean-Pierre said.
“President Biden has long been clear that he supports selling F 16s … to Turkey, which would help facilitate Nato interoperability.”