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We consider the asymptotic expansion of the generalised exponential integralinvolving the Mittag-Leffler function introduced recently by Mainardi andMasina [{\it Fract. Calc. Appl. Anal.} {\bf 21} (2018) 1156--1169]. We extendthe definition of this function using the two-parameter Mittag-Lefflerfunction. The expansions of the similarly extended sine and cosine integralsare also discussed. Numerical examples are presented to illustrate the accuracyof each type of expansion obtained.

Asymptotic expansion of the modified exponential integral involving the Mittag-Leffler function