General integral transform of the exponential integralsEn is considered and will be denoted asB(k)n(τ). Different expressions and the equations satisfied byB(k)n are developed. Two-term recurrence formula forB(k)n(0) and three-term recurrence formula forB(k)n(τ); τ≠0 will be established for a givenk≥1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn≥2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy