The announcement came after a meeting between Zimbabwe Cricket Union (ZCU) officials and representatives of the game's governing body in Dubai.
Zimbabwe had four Tests scheduled in the remainder of 2004, two in Pakistan in October and two at home to England in November.
But the decision does not cover One-day Internationals and England's continuing crisis of conscience over whether or not to tour has not abated.
The England and Wales Cricket Board (ECB) did, however, welcome the ZCU's decision.
"We greet that news with a degree of relief," ECB chairman David Morgan told BBC Sport.
"We are committed to playing four One-day Internationals, and I think there's a logic to it.
"Our Board will consider its position after (outgoing chief executive) Tim Lamb and I have reported on the ICC meetings which are due to take place in London and Monaco later this month."
The Pakistan Cricket Board (PCB) also welcomed the Zimbabwe decision.
"In the end it was a fair decision for Zimbabwe and now we have to play six One-day games or pursue one more team to play a tri-series One-day tournament when we host Zimbabwe in October," PCB chief executive Ramiz Raja said.
The ZCU offer comes after 15 senior Zimbabwe players went on strike following the sacking of the captain Heath Streak.
Streak had objected to the selection policies of the ZCU which he claimed were compromising the success of the national team.
Australia's two-Test series in Zimbabwe was cancelled last month because of the weakness of the opposition although they did play three One-dayers.
"Today's meeting demonstrates the depth of concern in the international cricket community over the impact the dispute in Zimbabwe is having on the game," said ICC president Ehsan Mani.
"These proposals will now go forward to the ICC executive board meeting on June 30 where the directors will be asked to support these actions.
"In the meantime, I've been assured by the ZCU that it will be prepared to take the players back without any conditions and the ICC remains hopeful that the ZCU and the players can resolve their differences."
The ICC will appoint a troubleshooter to investigate the claims of racism in the game in Zimbabwe.
India and South Africa will undertake 'A' team tours of Zimbabwe during July and August to expose the Zimbabwe team to more first-class cricket.
England spinner Ashley Giles, playing in the third Test against New Zealand at Trent Bridge in Nottingham, said : "We've just come off the pitch. We're not here to make decisions on Zimbabwe.
"We'll just have to wait and see what our Board says. It's out of our hands. We're just cricketers."
Meanwhile in London, a Foreign Office spokesman applauded the ZCU.
"We feel this is the right outcome. On this, as on all issues relating to Zimbabwe, it is far more effective to work for international action.
"It suits (Zimbabwe President) Robert Mugabe to present these issues as bilateral disputes with Britain.
"But it is far better to work internationally, as with European Union (EU) sanctions and the suspension from the Commonwealth."