NATO member Turkey was a key regional ally of Israel until the two countries cut ties in 2010 over the deadly storming by Israeli commandos of a Turkish aid ship bound for Gaza, which left 10 Turkish activists dead.
After years of bitter accusations and inflammatory rhetoric the two sides held secret talks in December to seek a rapprochement, with another round taking place in February in Geneva.
"The teams made progress towards finalising the agreement and closing the gaps, and agreed that the deal will be finalised in the next meeting which will be convened very soon," the Turkish ministry said in a statement after the fresh round of talks Thursday.
An Israeli official told AFP on condition of anonymity that "the gaps are getting narrowed," without elaborating.
Ankara said the latest talks in London brought together powerful Turkish foreign ministry official Feridun Sinirlioglu, Joseph Ciechanover, an advisor to Israeli Prime Minister Benjamin Netanyahu, and Israeli National Security Advisor Jacob Nagel.
In an interview with Turkish newspaper Vatan on Friday, Prime Minister Ahmet Davutoglu said "significant progress has been made", adding that the two sides were hoping to reach "a final stage" in the talks soon.
Russia and gas needs drive talks
Turkey had already said in February that the former allies were "close to concluding a deal".
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But they had yet to agree on all of Turkey's conditions, with the main hurdle appearing to be the lifting of Israel's blockade on the Hamas-ruled Gaza Strip.
Israel apologised to Turkey in 2013, in what many thought would trigger a warming in ties, but tensions soared again the following year when Israel launched a massive military offensive in Gaza.
Turkey's other condition is that victims of the Mavi Marmara be compensated, and has said that talks have advanced in this regard.
The Mavi Marmara was one of six ships which headed to Gaza in May 2010 carrying humanitarian supplies in a bid to break the blockade of the Strip which has been in place since 2007, shortly after the Islamist militant group Hamas won elections there.
The blockade has been branded cruel and inhumane by rights groups, although Israel argues it is necessary to stop weapons smuggling.
Turkey is a key backer of Hamas, and Erdogan has accused Israel of being "genocidal" in Gaza.
Analysts have suggested that Turkey's desire for a rapprochement has been accelerated by the drastic worsening in ties with Moscow since the shooting-down of a Russian warplane wrecked several joint projects.
Ankara relies on Russia for more than half its natural gas imports and Turkey now has its eyes on Israeli gas reserves.
In a highly symbolic encounter, Erdogan last week met representatives of Jewish organisations in the US to discuss the fight against terrorism and racism.
"Unfortunately, anti-Semitic, Islamophobic, xenophobic movements have been shifting from the periphery to the centre of politics. We have to struggle against them together," he said.
Another sign of a thaw in relations came when Israeli President Reuven Rivlin phoned Erdogan to thank him for his compassion after a suicide bombing in Istanbul last month left three Israelis dead.