According to officials in both countries, the disaster has killed at least 44 people. Some towns have been completely cut off and rescue teams feared the worst as improving weather allowed them to move in.
Some 10,000 people have been evacuated so far from the worst affected areas of northern Bosnia. In the town of Samac, hundreds of stranded residents were waiting to be rescued.
The worst rainfall since records began in the late 19th century caused landslides that brought more destruction and also prompted a landmine warning.
Bosnia's demining agency said residents around the towns of Doboj, Maglaj and Olovo -- which saw fierce fighting during the war in the 1990s -- should be particularly wary.
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Twenty of the 27 deaths recorded in Bosnia occurred in Doboj while on the other side of the Sava river, in Serbia, at least 16 bodies were found.
"Unfortunately there are estimates that the death toll will be higher," he said.
Croatia has also confirmed one casualty, as a result of bad weather that engulfed much of central and parts of eastern Europe at the weekend.
Ukraine emergencies ministry said electricity had been cut to about 100 villages in the north and west because of heavy rains and winds but that no casualties had been reported.
In the Serbian capital Belgrade, dozens of schools and sport centres were turned into shelters for the thousands of evacuees plucked the disaster zones by boats, buses and helicopters.
Mayor Sinisa Mali said he was confident any swelling of the Danube tributary could be contained but there was still concern further up river.