Jensen's covering theorem states that if $0^\sharp$ doesn't exist, then every uncountable set of ordinals can be covered by a constructible set of the same cardinality.

Now consider the following (somewhat) dual statements:

- Every uncountable set of ordinals covers a constructible uncountable set of ordinals.
- Every uncountable set of ordinals covers a constructible set of ordinals of the same cardinality.

I have two questions:

- Does any of the above statements follow from the non-existence of $0^\sharp$?
- If the answer is "no", are they still known to be consistent (in order to avoid trivialities, we may assume that V=L doesn't hold)?