"This visit could open a new chapter in relations between Turkey and Israel," Mr Erdogan told Turkey's NTV channel.
He said he was "ready to take steps in Israel's direction in all areas, including natural gas".
Relations between majority-Muslim Turkey and Israel froze after the death of 10 civilians in an Israeli raid on a Turkish flotilla carrying aid for the Gaza Strip in 2010.
But in recent months the two countries have been working on a rapprochement, with Mr Erdogan, a vocal supporter of the Palestinian cause, holding phone conversations with Mr Herzog and other Israeli leaders.
The Turkish president said last week that he was prepared to work with Israel on reviving an old project to ship Mediterranean gas to European clients through Turkey.
His remarks came after the US reportedly ended support for a rival pipeline project involving Israel and Turkey's historic rival, Greece.
Turkey had strongly opposed the project, which was supported by the former administration of US president Donald Trump.