A sequential estimation problem on a linearly combined parameter η := aθ+ bξ for a family of distributions with two parameters θ and ξ is considered. The Bhattacharyya type bound for the asymptotic variance of sequential estimators of η in the presence of a linearly combined nuisance parameter is given up to the second order and its attainment is also discussed. Further it is shown that the modified maximum likelihood estimation procedure of η with an appropriate stopping rule attains the bound for a family of normal distributions.