internship eligibility

Hello everybody!!! I have a question to ask and will be very grateful to however replies!

Here 's the story:

Graduated fron non-target and got an offer for the MSc Accouning and Finance at the London School of Economics starting this September(2016).I lost the offer due to a loan being approved late and have reinstated my application for the September 2017 intake.

Question: Although I have not yet received an offer for next year(September 2017) but will be studying anyway in the UK (I plan to study at any university if I do not get an offer from LSE again) am I eligible to apply for internships this summer(June 2017) BEFORE my Masters starts?

P.S.: I have been looking extensively at websites of banks but most of them focus on penultimate year students

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