Trump did provide written testimony on "certain" Russia-related topics, but not on "obstruction topics," said the report.
Democrats have demanded the full report, which Barr has said is redacted to protect grand jury material, sensitive intelligence, matters that could affect ongoing investigations and damage to the privacy rights of third parties, be made public to get a clearer picture of Mueller's investigation and conduct Congressional oversight.
Adam Schiff, chairman of the House Intelligence Committee, said Thursday that his panel has formally invited Mueller to testify on the investigation.
"The public deserve the facts, not Attorney General's political spin," the California Democrat tweeted.
Barr said Thursday that he has "no objection" to Mueller testifying before Congress about his investigation.
Mueller took over the investigation after Trump abruptly fired former FBI Director James Comey, a move that raised questions about his potential obstruction of justice.
The wide-ranging inquiry led to felony charges against 34 people, including six Trump associates and advisers, and three entities.
According to Mueller's report, the special counsel referred 14 investigations to other U.S. attorney's offices.
Russia has denied any meddling in the U.S. election.