Looking on the internet deeply has found these results:

do an internship is the most popular phrase on the web.

pursue an internship 172,600 results on the web	do an internship 1404,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
It doesn't look like this phrase is very popular! The search could not find any examples on the internet.	Took a year off to do an internship at the Sunnydale Mental Hospital. Groups of young executives are taught the language (for 9 months in Japan, a shorter period in Korea) and then do an internship in a company. I have been offering many young people for years the chance to do an internship in my MEP offices in Brussels or Strasbourg. I am therefore in favour of promoting the status of internships. You remember when I graduated from college and I went straight to Washington to do an internship?

pursue an internship

172,600 results on the web

do an internship

1404,000 results on the web

More popular!

It doesn't look like this phrase is very popular! The search could not find any examples on the internet.

Took a year off to do an internship at the Sunnydale Mental Hospital.
Groups of young executives are taught the language (for 9 months in Japan, a shorter period in Korea) and then do an internship in a company.
I have been offering many young people for years the chance to do an internship in my MEP offices in Brussels or Strasbourg. I am therefore in favour of promoting the status of internships.
You remember when I graduated from college and I went straight to Washington to do an internship?