Arriva said it was "in advanced discussion with Deutsche Bahn about the terms of a possible recommended cash offer" worth 775 pence per Arriva share.
Deutsche Bahn had informally offered 700 pence a share - or 1.2 billion pounds - in March, according to sources cited by Dow Jones Newswires.
Arriva ranks among the top British transport companies with a presence in 12 European countries where it operates bus and metro systems.
Deutsche Bahn, wholly owned by the German state, is trying to expand its business as private sector competition erodes its home market share.
Deutsche Bahn already owns Chiltern Railways in Britain and last year it looked at National Express, which runs bus and rail services, but decided against making an offer.