According to the paper, Fink, a 43-year-old German, agreed terms overnight and as part of the deal the Bundesliga side will pay the Swiss around one million Swiss francs (€800,000).
Fink, at Basel since 2009, is set to take charge of Hamburg for Saturday's visit to Freiburg as he takes the reins from sporting director Frank Arnesen, standing in over the past month since the sacking of Michael Oenning.
Hamburg are currently rock bottom of the Bundesliga with just one win in eight games this campaign.
Fink comes after leading Basel to the Swiss championship in both 2010 and 2011, also capturing the Swiss Cup in 2010.
He is set to be replaced at Basel by Lucerne's Murat Yakin.