Chaos has broken out at Kabul airport this week as tens of thousands of Afghans try to flee the militants, who swept through the country largely unopposed by government security forces.
Sport was tightly controlled by the Islamic fundamentalist group during their first rule of the country in the 1990s, often viewed by the militants as a distraction from religious duties.
Women were completely banned from taking part.
However, Shinwari said he saw no threats to cricket from the movement.
"Cricket was not an issue during the Taliban regime before and it will not be an issue now. I don't remember any incident caused by the Taliban over cricket," he said.
Shinwari said he was unable to comment on the status of women's cricket but the situation would be clearer in the coming weeks.
Star spin bowler and T20 captain Rashid Khan and all rounder Mohammad Nabi are currently playing The Hundred tournament in England. Both made pleas for peace in their country before the Taliban takeover.
But most other national players are in Afghanistan.
Sri Lanka's cricket board earlier this week said it was still hoping to host Afghanistan and Pakistan for the three matches in an empty stadium at Hambantota.
The one-day international series starts on 3 September.
It was shifted to Sri Lanka after stadiums in the United Arab Emirates -- where Afghanistan play their home matches -- were instead preparing to host the Indian Premier League.
The Afghanistan Cricket Board also announced its Twenty20 league, to start from 10 September in Kabul.
"We are committed to doing well and lifting Afghanistan cricket. We have excellent relations with Pakistan and Indian cricket boards and we are part of the international cricket community," it said.