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The “Due Process” policy in contract award was introduced to ensure the selection of the most appropriate contractor to deliver construction projects as specified so that the best value for money is ensured. Inspite of this, project delivery in Nigeria has been characterised by inefficiency in terms of cost overrun, time delay and poor quality standards. This study investigates the criteria often considered by clients in the award of contract after the prequalification exercise by a survey of questionnaire addressed to construction professionals in the Client, Contracting as well as Consulting organisations. Data collected were analysed with the aid of simple percentiles and mean score while the test of hypothesis was carried out using students’ t-test. The result reveals that the overriding criterion usually adopted for contract award after contractors’ prequalification still remains the lowest bidder because the client who has the final decision as to the award of contract cannot distinguish between lowest realistic bidder and lowest bidder. This has accounted for the low performance of projects procured under “Due Process” policy in terms of quality and time. The finding of this research is therefore a bold step and necessary guide for awarding contracts to the optimum tenderer whose tender figure is realistic enough to execute the project to the required quality within acceptable contract duration. It will further improve the effectiveness of the “Due Process” policy in construction project delivery in Nigeria most especially in the areas of time and quality performance.