Select permission inside stored procedure?
Ok, on the basis of the above comment and as per my suspicion - it seems as though you are trying to execute dynamic SQL within your stored procedure. 

What you need to remember is that when you do this it does not get executed within the context of the stored procedure - it gets executed within a new session. Because of this, the fact that the statement is being called within a stored procedure is a moot point, and you will need to grant explicit permission on the objects that your dynamic SQL is using. 

If you don't want to do this I would refactor your stored procedure to not use dynamic SQL.

The below link from Microsoft should help you with your problem:

PRB: Security Context of Dynamic SQL Statements Inside a Stored Procedure (Wayback Machine archive)


  This behavior occurs because a dynamic execution query (sp_executesql or EXECUTE) executes in a separate context from the main stored procedure; it executes in the security context of the user that executes the stored procedure and not in the security context of the owner of the stored procedure.


This is also discussed in the (more current) Microsoft Docs article:

Writing Secure Dynamic SQL in SQL Server


  Executing dynamically created SQL statements in your procedural code breaks the ownership chain, causing SQL Server to check the permissions of the caller against the objects being accessed by the dynamic SQL.