The studies of the last decade allow us to look in a new way at the possible association between the aging processes and the circadian rhythm. One of the promising directions in this area appeared thanks to the new data on the involvement of the NAD+-dependent protein deacetylase SIRT1 in the integration of the pathways of the circadian rhythm and metabolism regulation and the new NAD+ function as a “metabolic oscillator”. We present the a modification and extension of the most detailed circadian oscillator (CO) model developed in 2012 by J.K. Kim and D.B. Forger. The additional oscillator feedback with the involvement of NAMPT, SIRT1 genes/proteins, as well as NAM and NAD+, is included in it. The involvement of the CLOCK/BMAL1 transcription factor in regulating the NAMPT gene transcription determines the appropriate rhythm of mRNA expression and the NAMPT protein. Since the enzyme (this gene product) is a key in the pathway of NAD+ biosynthesis and recycling, the circadian rhythm is also typical for the fluctuation of this coenzyme level and the activity of NAD+-dependent protein deacetylase SIRT1. The deacetylation of the CO components by this enzyme closes the feedback mediated by this pathway. In particular, an increase in the Per2 protein degradation, an increase in the Bmal1 gene transcription, and the deacetylation of the chromatin of the regulatory regions of the CO genes in the area of E-boxes with subsequent transcription suppression can be revealed among the SIRT1 effects in the CO. All these processes are presented in the extended CO model that we suggested. Based on the experimental data about changes in the SIRT1 activity and NAD+ level with age, an attempt to study the effect of these age-related changes on the functioning of a CO was made. The modeling data indicate a decrease in the expression level of a number of CO genes (particularly the Bmal1 and Per2) in older age groups. An increase in the period of circadian oscillations was also registered. The results obtained indicate that a decrease in the SIRT1 activity associated with the age-related violation of the NAD+ metabolism can be one of the reasons for the violations of the functioning of a CO in the suprachiasmatic nuclei (SCN). Such violations can also entail violations of the organism’s circadian rhythms in general.