〜んばかり(に・の)
as if about to, almost
Verb ない-stem + んばかり(に / の / だ) (する → せんばかり)
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んばかり means something is so intense it looks as if it's about to happen: 彼女は泣き出さんばかりの顔をしていた, "she looked as if she was about to burst into tears".
It attaches to the ない-stem: 泣き出す → 泣き出さ + んばかり, 割れる → 割れ + んばかり. The ん is the classical volitional む. It's usually followed by の + noun, or に + verb.
Set phrases include 割れんばかりの拍手 ("thunderous applause"), あふれんばかりの笑顔 ("a beaming smile") and 飛び上がらんばかりに喜ぶ ("jump for joy").
言わんばかりに means "as if to say", and とばかりに (N1) is the other way to say it. Plain ばかり means "only".
Examples
彼女は泣き出さんばかりの顔をしていた。
She looked as if she was about to burst into tears.
会場は割れんばかりの拍手に包まれた。
The hall was filled with thunderous applause.
彼は「お前が悪い」と言わんばかりに、私をにらんだ。
He glared at me as if to say it was my fault.
彼女はあふれんばかりの笑顔で迎えてくれた。
She greeted us with a beaming smile.
合格の知らせに、彼は飛び上がらんばかりに喜んだ。
He practically jumped for joy at the news that he'd passed.