I am now a master student and get Ph.D. offers(entry in Oct 2018) from several UK universities, but I am faced with a dilemma which troubles me a lot. I am expected to gain my master degree in Jan 2019 theoretically, but I hope to graduate earlier in June this year because of internship opportunities, so I talked with my master's supervisor about graduation in advance and he agreed with it. However, the regulations in my university have changed since last month and new regulations require one EI paper in order to graduate in advance. For me, publishing an EI paper is difficult in such a short time without any research funding and that means it's hard for me to gain my degree in advance and enter the UK universities. Can I defer my entry to Jan 2019? How much is the possibility to defer my Ph.D. degree if I'm a self-funded student?

Only your proposed supervisor can answer this I'm afraid. You will need to ask them.

I'm sure it will be fine, especially if you are self funded.