A modular downhole packer includes a valve body, which includes a threaded coupler adapted to couple to a tubular and a valve assembly. The packer includes a packer element assembly, which includes a first end coupler, the first end coupler mechanically coupled to the valve body; a packer element; and a second end coupler. The packer includes a floating seal element mechanically coupled to the second end coupler, the floating seal element including a seal extension. The packer includes a floating end anchor. The floating end anchor includes a compression seal housing, the compression seal housing including an outer sealing surface positioned within the seal extension of the second end coupler; a compression seal positioned within a seal pocket formed in the compression seal housing; and an anchor body, the anchor body threadedly coupled to the seal pocket of the compression seal housing, the anchor body abutting the compression seal.