In a lossless system, the causal and acausal Green's function for elastic waves can be retrieved by cross-correlating the elastic field at two positions. This field, composed of converging and diverging waves, is interpreted in the frame of a time-reversal process. In this work, the near-field effects on the spatio-temporal focusing of elastic waves are analyzed through the elastodynamic Green's function. Contrary to the scalar field case, the spatial focusing is not symmetric preserving the directivity pattern of a simple source. One important feature of the spatial asymmetry is its dependency on the Poisson ratio of the solid. Additionally, it is shown that the retrieval of the bulk wave speed values is affected by diffraction. The correction factor depends on the relative direction between the source and the observed field. Experimental verification of the analysis is carried out on the volume of a soft-solid. A low-frequency diffuse-like field is generated by random impacts at the sample's free surface. The displacement field is imaged using ultrasound by a standard speckle tracking technique. One important application of this work is in the estimation of the shear elastic modulus in soft biological tissues, whose quantification can be useful in non-invasive diagnosis of various diseases.