The paper cited Friedrich Merz, a former CDU rival whom Merkel sidelined after becoming party chairwoman in 2000, as saying the coalition deal was a "humiliation" for the Christian Democrats. The SPD then suffered its worst result of the post-war era in September's election.
Martin Schulz was told by his SPD party members that they would not back the deal to form a government if he took the job.
After the SPD's losses in the September 24 national election, Schulz announced that the SPD would go into coalition and that he wouldn't hold talks with Merkel.
A Forsa poll had indicated that nearly three-quarters of Germans thought it would be wrong for Schulz to become foreign minister.
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Chancellor Angela Merkel's conservatives and Germany's main center-left party are still haggling over a deal to form a new coalition government after negotiations that lasted through the night. Schulz had suggested his successor should be parliamentary group chief Andrea Nahles, according to German media.
Chancellor Angela Merkel receives Valentine's Day flowers from the Central Gardening Association at the Chancellery in Berlin, Germany, February 9, 2018.
The SPD's 460,000 members will have the final say on the whole agreement in a postal vote, the result of which will be announced on 2 March.
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Kuehnert, 28, is travelling around Germany urging members to vote against the deal.
It's been over four months since its latest elections, and Germany finally has a government again.
Discontent also simmered in parts of the conservative bloc on Friday.
The SPD's traditional left-wing youth organization, Jusos, have opposed a coalition government with the Christian Democrats, and called for reform within the party.
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For the SPD, the backbiting went public when Gabriel, Germany's acting foreign minister, attacked the leadership for planning to remove him from his post - and implying that he'd been promised that he could continue as the country's top diplomat.