Nucleopeptides (NPs) represent synthetic polymers created by attaching nucleobases to the side chains of amino acid residues within peptides. These compounds amalgamate the characteristics of peptides and nucleic acids, showcasing a unique ability to recognize RNA structures. In this study, we present the design and synthesis of Fmoc-protected nucleobase amino acids (1,4-TzlNBAs) and a new class of NPs, where canonical nucleobases are affixed to the side chain of L-homoalanine (Hal) through a 1,4-linked-1,2,3-triazole (HalTzl). Fmoc-protected 1,4-TzlNBAs suitable for HalTzl synthesis were obtained via Cu(I)-catalyzed azide–alkyne cycloaddition (CuAAC) conjugation of Fmoc-L-azidohomoalanine (Fmoc-Aha) and N1- or N9-propargylated nucleobases or their derivatives. Following this, two trinucleopeptides, HalTzlAAA and HalTzlAGA, and the hexanucleopeptide HalTzlTCCCAG, designed to complement bulge and outer loop structures of TAR (trans-activation response element) RNA HIV-1, were synthesized using the classical solid-phase peptide synthesis (SPPS) protocol. The binding between HalTzls and fluorescently labeled 5′-(FAM(6))-TAR UCU and UUU mutant was characterized using circular dichroism (CD) and fluorescence spectroscopy. CD results confirmed the binding of HalTzls to TAR RNA, which was evident by a decrease in ellipticity band intensity around 265 nm during complexation. CD thermal denaturation studies indicated a relatively modest effect of complexation on the stability of TAR RNA structure. The binding of HalTzls at an equimolar ratio only marginally increased the melting temperature (Tm) of the TAR RNA structure, with an increment of less than 2 °C in most cases. Fluorescence spectroscopy revealed that HalTzlAAA and HalTzlAGA, complementary to UUU or UCU bulges, respectively, exhibited disparate affinities for the TAR RNA structure (with Kd ≈ 30 and 256 µM, respectively). Hexamer HalTzlTCCCAG, binding to the outer loop of TARUCU, demonstrated a moderate affinity with Kd ≈ 38 µM. This study demonstrates that newly designed HalTzls effectively bind the TAR RNA structure, presenting a potential new class of RNA binders and may be a promising scaffold for the development of a new class of antiviral drugs.