In terms of "is it possible", it's highly unusual to ask you to notify your current company before they offer you a job, for the obvious reason that it puts all the risk on the potential candidate, with nothing offered in return.
However, in the absence of specific labour laws and with a hiring culture that is not concerned with missing out on good candidates, a company can pretty much ask that you turn up in a frilly dress and refer to yourself as "Mrs Buttercup" for the duration of the interview. It's up to you if you want to jump through those hoops.
Realistically, you have three ethical options:
- Apply but explain politely that your company's policy does not support giving explicit permission to apply to other jobs (which is completely true, as there's no policy in place to support this)
- Get the permission
- Don't apply
Which one you choose depends on how comfortable you are with your current company, how much this new job would mean to you, and whether you are generally looking for a job.
If this job is particularly appealing, and you're sure your current employer can't match it, then there's nothing wrong with saying "I'm happy working here, but this is a tremendous opportunity, and I think it would be a really good fit for me". How well you think this would be received is entirely down to what the culture of the company you work for is; also how you fit into that company (for example, how high up you are, what other options you have, and how good your relationship with the decision-makers is)
If you're just generally looking, then my advice would be to pass on this job offer until you have a different one lined up and are in that "not currently employed but starting a new job" phase, at which point you can apply during the period that you don't have an employer.
One sensible question to ask is why they're asking for this permission, either directly, or by making contact with one of the employees of the bank who'd be expected to know.
It's possible they've been burnt by employers refusing to release employees from their contracts, and this is their attempt to save themselves some time and confusion. In which case, if you can state authoritatively that your company can't lock you into your current contract, then that should calm their concerns. You can even phrase it as "my company is willing to release me from my contract 1 month after we agree my start date with your company", which again is entirely true, because that's what the notice period in a contract means.
Of course, sometimes the rule is just the rule that HR always follows, and in that case, either you get the permission from your company, or don't apply. Other opportunities are always available.
One last thing, from some of the comments, I would advise you to seek out successful mentors in your own field and country. My experience of the UK/US hiring culture, with its relatively strong labour laws and acceptance that people look for the job that's right for them, may simply not be applicable to your situation; for all I know, it's culturally seen as an insult for you to be looking for work.
A senior employee of your current company would be more likely to know or to be experienced in what the reaction from management would be if you started looking into this, and asking them politely about that over lunch would be entirely appropriate, especially if you show them the job description.