We investigate the tachyonic cosmological potential V () in two different cases of the quasi-exponential expansion of universe and dis- cuss various forms of interaction between the two components—matter and the cosmological constant— of the tachyonic scalar field, which leads to the viable solutions of their respective energy densities. The distinction among the interaction forms is shown to appear in the Om(x) diagnostic. Further, the role of the high- and low-redshift ob- servations of the Hubble parameter is discussed to determine the pro- proportionality constants and hence the correct form of matter–cosmological constant interaction.