News agency dpa cited unidentified party sources who said that she is prepared to step down as party leader but remain as chancellor for now.
German Chancellor and leader of the Christian Democratic Union party, Angela Merkel, closes her eyes while speaking after exploratory talks on forming a new government broke down on 19 November 2017 in Berlin.
Merkel, 64, has been CDU chairwoman since 2000 and giving up the role would start a race within the party to succeed her as chancellor.
Political analysts in Germany have wondered whether Merkel might not serve out a full term if she's not leading her own party - a pairing that Merkel herself has said is crucial to being an effective chancellor.
"With these latest results, it has simply become untenable that Merkel continues to lead the CDU", said Mujtaba Rahman, managing director at Eurasia Group, a consultancy.
The exit poll for broadcaster ARD showed only 13 percent of CDU voters believed Merkel had helped the party in Hesse, down from 70 percent at the last state election, reflecting voter anger at her decision in 2015 to welcome nearly one million, mainly Muslim asylum seekers.
"Secondly, this fourth term is my last as German chancellor".
In Hesse, both her centre-right CDU and its coalition partners, the Social Democrats, were 10% down on the previous poll there - even though they remain in power. Merkel's party managed an unimpressive win, narrowly salvaging a majority for its regional governing coalition with the Greens.More news: Duchess Meghan sells out yet another dress
The move is not unprecedented in German politics.
The debacle followed a battering in a state election in Bavaria two weeks ago for the CSU and the Social Democrats.
In last year's general election Ms Merkel saw a drop in support and was forced to form a coalition with the centre-left Social Democrats following an inconclusive result. But there is precedent for splitting the two jobs.
Volker Bouffier, lead candidate for the German Christian Democrats (CDU), fills in his ballot paper in Hesse state elections on October 28, 2018 in Giessen, Germany.
Hesse's conservative governor, Volker Bouffier, told supporters that "the message this evening to the parties in the government in Berlin is clear: people want less argument, more objectivity, more solutions".
A new CDU chair will be elected at the upcoming party conference in December, Deutsche Welle said. However, the CDU's share of the vote fell by more than 11 points, to 27% - marking the worst result for the party since 1962.
Merkel's 13 years as chancellor have piled up baggage from repeated compromise-laden "grand coalitions" with the SPD, as well as a fateful 2015 decision to keep Germany's borders open, ultimately allowing in more than one million migrants.