The answer is simple,you need to change it back to old version and make it default, then try the integrate link in AdSense again.
I got a little frustrated today trying to integrate my Google AdSense account with Analytics, because when I would click on the Integrate with Google Analytics I wouldn’t get the option to create new account or connect with an existing account, I would just get redirected to my Analytics home page.
So if you are using the new interface version on your Analytics account try this.
In the upper right corner on your Analytics page select the old version link followed by the Make this version default link and then try clicking in the integrate link on you AdSense account.
UPDATE TO POST
(You can only have your AdSense linked to one Analytics account, even if they are all on the same Google account)
To answer the question in the comments.
What if you have ‘View performance in Google Analytics’ instead of the ‘Integrate with Google Analytics’.
That means that your accounts are already linked, if you don’t see it on your domain Analytics page, in AdSense Overview, if you get the message:
This report requires AdSense to be enabled for this profile.
What is AdSense?
Google AdSense is a free, simple way for you to earn money by displaying targeted Google ads on your website. AdSense also lets you provide Google search to your site visitors, and earn money by displaying Google ads on the search-results pages.
How does linking my AdSense account to my Analytics account help me?
When you link your AdSense account to your Analytics account, you can see key AdSense metrics (e.g., eCPM, Unit Impressions) correlated with Analytics metrics.Learn more
then your AdSense is linked to a different domain.
Some important facts if you want to have AdSense on multiple domains:
You can only link your AdSense account to one Analytics account but multiple profiles. Take a look at the picture below to see difference between profiles and accounts
I have two accounts here Admin Ramble and Beginner Blog, but in Admin Ramble account I have two profiles.
If I want to track AdSense earning on both adminramble.com and beginner.adminramble.com I have to make beginner.adminramble.com as a different profile inside the same account as adminramble.com.
I’m working with a subdomain name here, but the principle is the same if you have two totally different domains.
If you want to track multiple domains you will have to add a Code Snippet above the Analytics tracking code. The primary domain code doesn’t need to be changed.
How to see which to which account is AdSense linked, which domain is primary and what code snippet to add?
Here is the procedure for both old and new version interface.
OLD VERSION interface
Go to Analytics Settings in the upper left corner of the page under the Google logo, and select one of your accounts.
If the account you selected is linked with AdSense you will have ‘Edit AdSense linking settings’ link
If the link is missing, you only have ‘Edit account settings’ link , then that is not the linked account
Go to the account which is linked to AdSense and click on ‘Edit account settings’ link, select the domains you wish to link and click ‘Continue’.
Select the domain you wish to be the primary and click ‘Continue’
On the next page you will be represented with the code you have to paste above the Google Analytics tracking code
NEW VERSION interface
Go to ‘All Acounts’ and ‘ACCOUNTS LIST’ under the Google logo in the upper left corner of the page
and then on the Accounts Home page go to Admin section in the right upper corner
You will be taken to Account Administration page
There select one of you accounts and then go to ‘Data Sources’ and then ‘AdSense’ tab
There you will see which domain is the primary one and links to get code snippet to add to your page, to make some other domain primary or to unlink your AdSense account.