Is it acceptable to propose the same PhD research plan to two universities? Well I have proposed a research(starting next year) and accepted by supervisor and university (formally applied and accepted by university). I have a conditional offer letter. But I need funding for it so I have applied for an external scholarship(good chances but not 100% sure) but the decision of scholarship will be made only after 3-4 months. If I don’t get scholarship or funding, I think, I won’t be able to pursue the research or study.

And, there is another university (but 2nd preference) with some very good internal scholarship options and I have very good chances looking at their criteria. I also have a supervisor, whom I know academically, who is ready to supervise the research from that university. But off course he cannot make decision on scholarships. I am thinking of applying to that university as a backup so that I don’t lose one year waiting next year’s scholarship round if I don’t get the first scholarship. And also, after 3 months there won't be any scholarships deadline left for this year.

So, is it good thing to apply for same research project in two different universities? (Off course, ideally, it is not but in my case what should I do?).

And are we allowed to do so? Is it normal? (Off course, finally I will have to select one between two).

I am afraid that after all the hassle and correspondence with both the future supervisors, how I may be able to deny one, if I am selected for the both.

And also if I say to second university’s supervisor that I have already applied in another university as well and only applying as backup, firstly it may be rude (who knows, may have ego problems later) , secondly, that may even diminish my chances in 2nd university as well(admission and scholarship).

So, I am in dilemma what I should do? Please suggest.
Chances are that your supervisor at the first university is sensitive to the challenges in acquiring funding. If you have been clear to him/her that you cannot pursue the studies without support and are solely dependent on external funding, then he/she should certainly understand if you pursue them elsewhere where funding is available. Furthermore, if you are up front about it, the supervisor may scramble more to find funding for you. The situation is not unlike courting two potential employers during a job search.

I would not recommend suggesting to the second supervisor that he/she is a backup option, but only that you have also applied with a warm reception to the first university. This way, you do not risk stepping on egos, but still make it clear that you are entertaining other options.