Issue:

Outlook get disconnected after o365 migration.

Solution:

Option 1 - Update your Office and software to the latest version

Option 2 - Reset/Remove your account credentials from within your OS

            Try going into the control panel>Credential Manager and clearing any cached credentials.

Option 3 - Modify the registry for modern authentication

                    Create the following registry key in order to force Outlook to use the newer authentication method for web services, such as EWS and Autodiscover.

  1. Exit Outlook.

  2. Start Registry Editor. To do this, use one of the following procedures, as appropriate for your version of Windows:

  3. In Registry Editor, locate and then click the following registry subkey: 

    HKEY_CURRENT_USER\Software\Microsoft\Exchange

  4. On the Edit menu, point to New, and then click DWORD Value.

  5. Type AlwaysUseMSOAuthForAutoDiscover, and then press Enter.

  6. Right-click AlwaysUseMSOAuthForAutoDiscover, and then click Modify.

  7. In the Value data box, type 1, and then click OK.

  8. Exit Registry Editor. 

Option 4 - To fix this issue, change the Logon network security setting to Anonymous Authentication.

Option 5: This is linked to option 4.

When attempting to create an Outlook profile or connect to an Office 365 mailbox, you are continually prompted for credentials while the client displays "trying to connect..." If you cancel the credentials prompt, you receive the following error:

The connection to Microsoft Exchange is unavailable. Outlook must be online or connected to complete this action.
Under More Settings on the Microsoft Exchange Security tab, the dropdown for Logon network security displays a value other than Anonymous Authentication, and is disabled.

Remove the AuthenticationService registry value, or disable the Group Policy which is applying it.

Using a registry

This behavior can occur if the AuthenticationService registry value is configured. This registry value will be found in the following location:
HKEY_CURRENT_USER\Software\Policies\Microsoft\Office\x.0\Outlook\Security 

DWORD: 
AuthenticationService
Available values: 
9, 16 or 10
Where x.0 = 
15.0 for Microsoft Outlook 2013, and x.0 = 14.0 for Microsoft Outlook 2010, and 12.0 for Microsoft Outlook 2007.

Using a group policy

Policy: Account Settings\Exchange\Authentication with Exchange Server 
Setting: 
"Select the authentication with Exchange server."

Available choices:
Kerberos/NTLM Password Authentication
 (Default) 
Kerberos Password Authentication
NTLM Password Authentication 

To disable the Group Policy, under 
Authentication with Exchange Server, select Not Configured.

Option 6: Disable any add-ons and test.

Option 7: Make sure your mailbox is not part of a shared mailbox in Exchange online or migrate the shared mailbox.

Option 8: download fix outlook problem

http://diagnostics.office.com/#/