In OpenGL it's possible even without shaders:
// Set color, your case
glTexEnvi( GL_TEXTURE_ENV, GL_TEXTURE_ENV_MODE, GL_COMBINE_ARB );
glTexEnvi( GL_TEXTURE_ENV, GL_COMBINE_RGB_ARB, GL_REPLACE );
glTexEnvi( GL_TEXTURE_ENV, GL_SOURCE0_RGB_ARB, GL_PRIMARY_COLOR_ARB );
// Mix colors
glTexEnvi( GL_TEXTURE_ENV, GL_TEXTURE_ENV_MODE, GL_MODULATE );
But I'm not sure if it's in any SFML functions.