If there is a Partnership Agreement in place the answer is most likely, yes. A Partnership Agreement governing the activities of the partnership and conduct of the partners will often place some restrictions on the nature of the interest which may be acquired.
For example, the transferring partner might be limited to transferring only their economic interests and rights which would prevent the recipient of transferred interest from becoming a full partner (with voting rights and managerial input) by assignment alone. Full admission to the partnership would be decided by the remaining partners based on the terms of the Partnership Agreement.
The category of assignee is something else the partnership might have good reason for restricting. For example, federal tax audit rules introduced in 2018 mean that partnerships will be treated as taxable entities if one or more of the partners is itself a partnership, a trust, or an LLC. To avoid such tax consequences, and preserve individual tax treatment for the partners, the partnership agreement might prohibit assignments of partnership interest may be sold to any such business entity.