Officials from both sides met in Haiti on Saturday as part of what they called “a process of rapprochement”.
The US and Venezuela have not had ambassadors in each other’s country since 2010.
Relations have worsened since the US earlier this year imposed sanctions on a number of Venezuelan officials it accuses of human rights abuses.
US President Barack Obama issued an executive order in March freezing the assets of seven Venezuelan officials and banning them from entering the UnitedStates.
He also said the situation in Venezuela, including “the government’s erosion of human rights guarantees (…) constitutes an unusual and extraordinary threat to the national security and foreign policy of the United States”.
The decree caused outrage not only among Venezuelan officials but also many Latin American leaders sensitive to what they see as US imperialist rhetoric.
The US later said that the wording was “completely pro-forma” and the US and Venezuelan presidents briefly spoke at a regional summit in Panama in April.
The newspaper quoted “people familiar with the probe” as saying that several high-ranking Venezuelan officials, including Mr Cabello, were being investigated on suspicion of having “turned the country into a global hub for cocaine trafficking andmoney laundering“.
Mr Cabello has denied the allegations and Venezuelan President Nicolas Maduro said that anyone who attacked the National Assembly chief was attacking him.
Mr Cabello was part of the Venezuelan delegation that met US State Department counsellor Thomas Shannon on Saturday.
He said that it was “an important step toward the full restoration of relations between the countries”.
The State Department on Monday also called the talks “positive and productive”.
President Maduro said the two sides had “opened a very important diplomatic channel”.