The government has been through two major crises, first over whether to turn back small numbers of migrants at the German-Austrian border and then over what to do with the head of Germany's domestic intelligence service after he was accused of downplaying far-right violence against migrants.
It was the second regional vote in quick succession to disappoint the conservative camp, after a battering in Bavaria two weeks ago.
Projections for ARD and ZDF public television, based on exit polls and partial counting, gave the CDU 27-28 per cent support and the centre-left Social Democrats almost 20 per cent.
The dismal performance poured further scorn over Merkel's authority as a leader of the CDU. She has been chancellor for 13 years.
Ms Merkel's predecessor, Gerhard Schroeder, stepped down as leader of his party in 2004 and remained chancellor for another 18 months.
Party sources said Merkel's favoured successor, CDU party secretary-general Annegret Kramp-Karrenbauer, had announced her candidacy for the party chair.
The result suggests the CDU and Greens could continue their ruling coalition in Hesse but is likely to increase tensions in Merkel's ruling "grand coalition" in Berlin.
Earlier in October, Merkel's ruling coalition was shaken after the Christian Social Union (CSU) - the sister party of the CDU - gained 37.3 percent in Germany's largest and second-most populous state of Bavaria.
Merkel, who has been CDU chairwoman since 2000, told a party leadership meeting of her offer on Monday morning, after the party reeled from heavy losses in two recent regional elections, most recently on Sunday in Hesse. Although the CDU has remained relatively united behind its leader in public, discontent has been building as the party's fortunes have sunk.
The CDU vote fell 11.3 per cent compared to the last elections in 2013.
Merkel is under pressure from her Social Democrat coalition partners to deliver more policy results and the centre-left party could yet pull out of the government at a mid-term review next year.
And Mr. Bouffier, noting that his party fared better in the Hesse vote than it now does in polls nationally, seems keen to stay in power.
Sunday's election in the central state of Hesse saw both Merkel's conservative Christian Democratic Union and the center-left Social Democrats lose significant ground, while there were gains for both the Greens and the far-right Alternative for Germany.
Merkel is now headed for a re-election bid in December within her own party, where only 13% of responding CDU voters believe that Merkel has helped the party within Heisse, and this marks the fifth time that Merkel's government has come close to collapsing in on itself from within.
Just five years after the party's founding as a protest against euro zone bailouts, the anti-immigrant AfD now controls seats in all 16 of Germany's state parliaments, in addition to the national Bundestag and the European parliament.