Legally, only the contracted person can buy themselves out. The new company cannot do that, as they have no contract with the current company.
So, in that respect, it is not so strange, and I would not treat it as a red flag.
Now, I know that being “caught in limbo” between having paid the money and getting it reimbursed makes you nervous – I would be too.
But, if you travelled for the interview with the new company, the same thing happened with travel expenses.
So, same principle, just – presumably – a much larger account.
You absolutely must (try to) get the new company to state in writing that they will reimburse.
If they won’t, then first ask why (and take it as a red flag, no matter what their answer). You can also hint, according to how much you want/need the new job, that you may not be able to accept if they do not give you a written undertaking to reimburse.
Also, consider the size of the new company. A large multinational might be “safer” than a 3 or 4 employee outfit (but there is no guarantee).
Tl;dr – it is natural to be nervous, but if you have visited the company premises for interview then you should be able to judge how much you trust them, and a written undertaking from them to reimburse you ought to be legally enforceable (IANAL).