The Senate's top Republican is warning that legislation enacted over President Barack Obama's veto to allow families of Sept. 11 victims to sue Saudi Arabia in U.S. courts may have "unintended ramifications" and that lawmakers should discuss fixes to the measure.

Majority Leader Mitch McConnell said the White House was too slow to warn about the "potential consequences" of the measure. Both the House and Senate overwhelmingly overrode Obama's veto of the measure on Wednesday.

Those concerned about the law warn that it could have a chilling effect on Saudi Arabia's cooperation with the U.S. in fighting terrorism. They also warn that it could trigger lawsuits from people in other countries seeking redress for injuries or deaths caused by military actions in which the U.S. may have had a role.