Shanghai and Suzhou pay social security.

Whether your household registration is in Suzhou or not, as long as you have paid urban social insurance and had a social security account in Suzhou.

Then, suppose you are working in Shanghai now, and after the Shanghai unit has handled the social security account in Shanghai for you, you can go to Suzhou Social Security Center to open the transfer-out form, and then give it to the person in charge of social security in the unit, so that he can go to the district-level social security center where the unit is located to handle the transfer-in. According to personal circumstances, you can use the original Suzhou social security payment details (you can go to Suzhou Social Security Center to print it out).

If you don't pay the city insurance in Suzhou, I suggest you go to Suzhou to apply for the city insurance first and make up a sum of money, which will be good for the future.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.