The scriptures show a special connection between love and the Holy Spirit. Examples of such exists in Romans 5:5, which says "And hope maketh not ashamed; because the love of God is shed abroad in our hearts by the Holy Ghost which is given unto us." and Galatians 5:22, which attributes love (and actions of love such as kindness) as fruits of the Spirit. These two texts show a special connection between love and the Holy Spirit, as it is through the Spirit that the love of God is shed to our hearts. Although these two texts are not enough to alone justify the classical western view, they do implicity suggest the view.
Another example of this special relation is found in the book of Samuel, where when the Holy Spirit departed from Saul (1 Samuel 16:14), it is said that the love of God left Saul "but my steadfast love will not depart from him, as I took it from Saul, whom I put away from before you" (2 Samuel 7:15). Thus, the author of Samuel seems to identify the Spirit with love. The New Testament also depicts the Holy Spirit with the dove (Luke 3:22) which is a symbol of love in Jewish culture ( Song of Solomon 6:9, Song of Solomon 2:14). The fact that He is called "gift" (Acts 2:38) also shows the Holy Spirit's relation to love, as gifts proceed from love.
These texts together do seem to suggest that the western view of the Spirit's procession is scriptural.