Erdoğan and the crown prince also agreed "to increase the efforts to end the tension in the axis of Qatar," during a telephone call, according to sources.
Erdoğan and Saudi King Salman bin Abdulaziz will have a face-to-face meeting at the G20 Summit on July 7-8 in Hamburg, Germany.
In a royal decree early Wednesday, Salman placed his son, Mohammed, 31, as first in line to the throne.
He also relieved Prince Mohammed bin Nayef, 57, from his position as the deputy prime minister and interior minister.
Saudi Arabia, Egypt, the United Arab Emirates, Bahrain and Yemen severed ties with Doha on June 5, and shut their air, land and sea borders to Qatari vessels earlier this month. Qatar's neighbors accuse it of supporting terror groups -- a claim Doha denies.
Mauritania followed suit shortly afterwards, while Jordan downgraded its diplomatic relations with Doha and closed the local office of Qatar's Al-Jazeera satellite news channel.
Qatar has denied the accusations and said the move to diplomatically isolate it was "unjustified".
That country's Emir Sheikh Tamim bin Hamad Al-Thani on Wednesday also congratulated the new crown prince, according to state news agency QNA.
In separate cables to the king and the prince, Doha said it wished for "more progress for brotherly relations between the two brotherly countries", QNA said.
