Recep Tayyip Erdogan and Vladimir Putin agreed over the phone to meet in person, Mr Erdogan's office said.
The call came after Turkey expressed "regret" earlier this week to Mr Putin and to the family of the Russian pilot killed in the incident.
The fighter jet was shot down near the Syria-Turkey border in November.
Turkey claimed that the jet was warned repeatedly after entering Turkish airspace, a claim fiercely denied by Russian officials.
In the call, Mr Putin also called an attack on Istanbul's Ataturk on Tuesday "heinous".
A spokesman for Mr Erdogan said on Wednesday that "necessary steps" would now be taken to mend the damaged ties between the two countries, and to drop restrictive measures on Russians visiting Turkey.
"Reiterating their commitment to reinvigorate bilateral relations and fight terrorism together, the two leaders agreed to remain in contact and meet in person," he said.
Turkey angered Moscow in the aftermath of the incident by failing to apologise, despite demands from Russian officials. Mr Putin said he had been stabbed in the back and accused Mr Erdogan of collaborating with so-called Islamic State.
Russia responding by hitting Turkey with a raft of sanctions, banning import of Turkish foodstuffs and stopping Russian package holidays to Turkey.
Mr Putin said at the time the restrictions would not be lifted until Russia received an apology.
The Russian Su-24, an all-weather attack aircraft, was flying in skies above the Turkey-Syria border area on 24 November when it was shot down by Turkish F-16s.
The plane crashed in the mountainous Jabal Turkmen area of the Syrian province of Latakia, killing the pilot.
A Russian marine involved in a helicopter rescue attempt was killed when the helicopter came under fire from local fighters.
The navigator in the jet, Capt Konstantin Murakhtin, survived the crash and was taken to Russia's Hmeimim airbase in Latakia by Syrian government forces.
Turkey said planes were warned 10 times during five minutes via an "emergency" channel and asked to change direction.
The Russian defence ministry insisted that the aircraft remained within Syria's borders throughout its mission and did not violate Turkish airspace and received no warnings.

The former Italy goalkeeper, 56, replaced Kenny Jackett on 30 July, shortly after Chinese conglomerate Fosun International bought the club.
Despite significant investment in the playing squad in August, Wolves have won only four of 14 league games and are 18th in the Championship.
Zenga's assistant, Stefano Cusin, has also left Molineux.
First-team coach Rob Edwards has been placed in temporary charge and will prepare the team for Saturday's game at Blackburn.
A 1-0 home defeat by Leeds on 22 October was Wolves' fourth loss in five league matches, although midfielder Dave Edwards told BBC WM 95.6 after the game that Zenga retained the backing of his playing squad.
Nine Championship clubs appointed new managers before the start of the 2016-17 season, but Zenga's departure from Wolves means only four of them remain in their posts.
Roberto di Matteo, Paul Trollope, Nigel Pearson and Alan Stubbs left Aston Villa, Cardiff, Derby and Rotherham respectively in October.
Rob Gurney, BBC WM 95.6
The cut-throat, "success yesterday" culture of football in the modern era is perfectly illustrated here, with Zenga given less than three months to prove himself.
His training methods were clearly popular with the players, but equally some of his personnel decisions were deemed curious.
Integrating a huge number of new players, however technically proficient, was not going to be easy, but owners Fosun have decided that someone else can do better in the pursuit of the boundless riches of the Premier League.
Zenga's passion and commitment was clear. He was often an adversarial interviewee, steadfastly believing that he was right and the inquisitor was wrong.
Ultimately his employers decided he was wrong. And so to the next…
The leaders of Turkey and Russia have spoken for the first time since the downing of a Russian military jet by Turkey sparked a diplomatic crisis.
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Championship club Wolves have parted company with head coach Walter Zenga after just 87 days in charge.