"We have thought of it. The idea has been floated with the BCCI if an Ashes like series can be played between India and Pakistan every year or alternate year," Ashraf told reporters.
"The series can be named either after ( Mohammad Ali) Jinnah or (Mahatma) Gandhi."
He said the level of understanding between both the countries, even at the Prime Minister level, has improved and the two governments are moving in a positive way.
"Even the trade relations have improved now. India has its own economic interest in this and so has Pakistan. It is mutually beneficial," the PCB chief said.
Ashraf said he is working with the BCCI President N Srinivasan to develop a very good working relationship and develop a better understanding.
"Hopefully in future we will move at a dramatically high speed. Wherever I go people ask me why can't we have more matches with India. Why should fans be deprived of it?" he questioned, adding that the ongoing bilateral series, resumed after a gap of five years, is a dream come true moment for him.
Hoping that the Indian cricket team will visit Pakistan soon, the PCB chief said future matches with India can be played in Pakistan or at a neutral venue.
He went on to say that no other cricket game can be bigger in the world than an Indo-Pak match.
"It has been my wish from day one that there should be revival of cricket (between India and Pakistan). It is good for cricket as well as the people," Ashraf said while stressing that such games enhance people-to-people contact as well as spread the message of peace and friendship.
On relations with other cricketing nations, the PCB chief said things are much better now.
"The terrorists have moved away to the northern side of the Afghanistan border. Our cities and grounds are now safe. We have proper security in place and no team should be scared of coming to Pakistan," he said.
Bangladesh had recently refused to tour Pakistan citing security concerns and Ashraf said their decision was influenced by the terrorists attack on the Sri Lankan cricket team in 2009.
"Old fear of that unfortunate incident in 2009 prevails. So, they are scared that such an incident might get repeated. But I want to assure everyone that now there is no threat," the PCB head said adding that Bangladesh should honour their commitment.
Ashraf said even the ICC is supporting that revival of cricket in Pakistan.
On the inclusion of Pakistani players in the sixth edition of the IPL, he said, "If the BCCI asks us, we will be cooperative and supportive as we have been in the past."