The Federal Office of Information Technology Security (BSI) decided that it could no longer defend against the attack and had to give up, broadcasters NDR and WDR and the Süddeutsche Zeitung reported on Thursday.
Der Spiegel reported on Wednesday evening that data was still flowing out of the parliament to unknown recipients, and that rebuilding the network using completely new hardware would take months and cost millions.
Bundestag spokespeople would not comment on the media reports on Wednesday evening, saying that it was up to the responsible committees of MPs to respond.
MPs are due to debate the parliament's next steps on Thursday, including whether to call on Germany's domestic intelligence service, the Federal Office for the Protection of the Constitution (Bundesamt für Verfassungsschutz) for help.
But some MPs have expressed doubts about allowing the executive-controlled security services access to their computers.
"An executive agency might look into the legislative branch – so you'd rather be spied on by a [foreign] secret service. It's crazy," CDU MP Armin Schuster said of his colleagues' fears to Spiegel Online.
Another MP told the news magazine that "given the dimensions of the hacker attack, what's going on here is dangerously naive".
News of the unprecedented attack became public knowledge a month ago, when it emerged that unknown hackers had smuggled a trojan (disguised computer virus) into the network and used the access it provided them to funnel off data.
While no-one can say with certainty who is reponsible, many have speculated that a foreign intelligence service could be behind the attack.
"Experts have clues that the Russian foreign intelligence service SWR is behind the espionage action," Der Spiegel wrote on Wednesday.