According to the source, Tehran expressed its understanding over the negotiations and is also holding talks "with other elements".
In a phone call with the newspaper, the source said that "serious progress has been made in the talks." He added that both sides were on the verge of launching direct talks, but that direct negotiations depended on developments in the diplomatic and regional conditions.
According to the source, the next round of talks is expected to deal with "the fundamental issues". He added that the regional issues, including the ties between Syria and Iran, Hizbullah and Hamas, were postponed to future talks.
On Tuesday, Syrian President Bashar Assad dismissed Israeli demands for Syria to abandon an alliance with Iran as a requirement for a peace deal.
Assad told British MPs that the Baath Party government intended to maintain its "normal relations" with Iran while it conducts indirect talks with Israel to regain the Golan Heights, a source familiar with the meeting told Reuters.
The source stressed that an agreement had been reached in the past regarding the main principles in four issues raised in the talks: The security arrangements, the water, the borders and the normalization.
The source, who spoke on condition of anonymity, Israel accepted the security arrangements between the two countries, which will be temporarily implemented until trust is built between them.
"The Syrians insist that the security arrangements will be shared by both sides," he said. "There is an Israeli agreement on this issue. This matter was dealt as a fundamental issue along with the rest of the issues discussed with the Israelis."
Reuters contributed to this report