From the introduction to the CRS report, Presidential Advisers’ Testimony Before Congressional Committees: An Overview (December 15, 2014, RL31351):
Since the beginning of the federal government, Presidents have called upon executive branch officials to provide them with advice regarding matters of policy and administration. While Cabinet members were among the first to play such a role, the creation of the Executive Office of the President (EOP) in 1939 and the various agencies located within that structure resulted in a large increase in the number and variety of presidential advisers. All senior staff members of the White House Office and the leaders of the various EOP agencies and instrumentalities could be said to serve as advisers to the President.
Occasionally, these executive branch officials playing a presidential advisory role have been called upon to testify before congressional committees and subcommittees. Sometimes, such invited appearances have been prompted by allegations of personal misconduct on the part of the official, but they have also included instances when accountability for policymaking and administrative or managerial actions have instigated the request for testimony. Because such appearances before congressional committees or subcommittees seemingly could result in demands for advice proffered to the President, or the disclosure—inadvertent or otherwise—of such advice, there has been resistance, from time to time, by the Chief Executive to allowing such testimony.
Congress has a constitutionally rooted right of access to the information it needs to perform its Article I legislative and oversight functions. Generally, a congressional committee with jurisdiction over the subject matter, which is conducting an authorized investigation for legislative or oversight purposes, has a right to information held by the executive branch in the absence of either a valid claim of constitutional privilege by the executive or a statutory provision whereby Congress has limited its constitutional right to information.
A congressional committee may request (informally or by a letter from the committee chair, perhaps co-signed by the ranking Member) or demand (pursuant to subpoena) the testimony of a presidential adviser. However, Congress may encounter legal and political problems in attempting to enforce a subpoena to a presidential adviser. Conflicts concerning congressional requests or demands for executive branch testimony or documents often involve extensive negotiations and may be resolved by some form of compromise as to, inter alia, the scope of the testimony or information to be provided to Congress.