IN THIS PAPER, WE INTRODUCE THE NOTION OF K-RANK NUMERICAL RANGE OF RECTANGULAR COMPLEX MATRICES. SOME ALGEBRAIC AND GEOMETRICAL PROPERTIES ARE INVESTIGATED. WE ALSO, FOR E, WE INTRODUCE THE NOTION OF BIRKHOFF-JAMES APPROXIMATE ORTHOGONALITY SETS FOR E-HIGHER RANK NUMERICAL RANGES OF RECTANGULAR MATRICES. THE PROPOSED DEFINITION YIELDS A NATURAL GENERALIZATION OF STANDARD HIGHER RANK NUMERICAL RANGE.