A classical construction of Katz gives a purely algebraic construction of Eisenstein-Kronecker series using the Gau beta-Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and p-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein-Kronecker series via the Poincare bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable p-adic Eisenstein measure through p-adic theta functions of the Poincare bundle.