Binali Yildirim's remarks came during a program televised live on state broadcaster TRT after Erdogan urged the restoration of “traditional friendly ties” in a letter to Putin.
"I suppose our president will have a phone call with Putin today, tomorrow, maybe Wednesday, maybe Thursday," Yildirim said.
"The content of the letter is very clear. We express our regret. We say that we are sharing the pain felt by those who lost loved ones. We say that we will pay compensation, if needed."
Yildirim also said bilateral relations should return to normal. "The other side approves and confirms it," he added.
Following the downing of a Russian warplane in Turkish airspace last November, the Kremlin ordered sanctions on food products, an end to visa-free travel and a ban on Russian tourists taking package holidays in Turkey.
Yildirim also announced that a planned agreement between Turkey and Israel to normalize ties, expected to lead to an improvement of the humanitarian situation in the Gaza Strip, will be formally signed Tuesday.
"This agreement took a long time. Our matter with Israel was apology and compensation, but our president did not narrow it down,” he said. “The fundamental center of gravity of the agreement is Palestine issue," he added.
Erdogan said late Monday that more than 10,000 tons of supplies will sail to Gaza from Turkey before the Eid al-Fitr religious holiday in early July, prompting Hamas to thank Erdogan for his efforts to ease the blockade.
The accord will pave the way for the restoration of bilateral relations more than six years after Israeli commandos stormed a Gaza-bound Turkish aid flotilla in international waters, killing nine Turkish activists and injuring another 30, one of whom succumbed to his injuries nearly four years later.
In the aftermath of the attack, Turkey demanded a formal apology from Israel, compensation for the families of those killed and the lifting of Israel’s blockade.
In 2013, Netanyahu voiced his regret for the attack.
On Monday, months of talks between the two countries finally bore fruit, with Turkey announcing that a deal would be signed the following day to normalize relations with Israel.
According to the deal, Israel will pay a total of $20 million in compensation to the families of the Mavi Marmara victims.
Turning to Egypt, Yildirim said relations with Cairo will begin at the ministerial level.
"It may happen. There is no obstacle. Actually, we are ready. We do not have any reservations," he said.
Asked whether a Turkish minister will go to Egypt or a meeting will be held, Yildirim said, "Mutual. Egypt [ministers] will come, ours will go. Businessmen may come, cultural exchanges are possible, military mutual contacts may be reached. These are all possible, there is no problem."
Although there was a military coup against democracy in Egypt, Yildirim added, "Let’s put it aside but on the other hand life goes on. We are living in the same region. We need ourselves."
Egypt's first democratically elected leader, Mohamed Morsi, was ousted by the military in a 2013 military coup following protests against his presidency.
He has since been handed life-in-prison and death sentences for "conspiring against Egypt" -- allegedly with Palestinian group Hamas and Lebanon’s Hezbollah -- and for escaping jail in 2011.
He has also been sentenced to a 20-year prison term for allegedly committing murder.
Morsi and his co-defendants, along with a number of independent observers, say the charges are politically driven.
Since Morsi's ouster and imprisonment, Egyptian authorities have launched a harsh crackdown on his Muslim Brotherhood group -- killing hundreds and jailing tens of thousands.