I believe that Lie Scale from EPQ-R measures exactly what you're looking for:

https://shop.acer.edu.au/acer-shop/group/RT

HI Bishakha,

It rather depends on the survey and what you want to do.

The various inventories such as BIDR measure an individual's presentations: self-deception and deliberate self-presentation.  If your research is an attempt to measure self-presentation, then such instruments can do that for you.

However, I am not sure how you would apply such inventories to a questionnaire on another topic, unless you administered the inventory and your other questions separately and then analysed the responses for each individual against their perceived presentation preferences (see for example Article Socially desirable responding in personality assessment: Not...

If you want some internal validation processes, then you might add something like the 'Lie' scale into your questionnaire as well. But once again it is, or rather it is part of, a personality inventory with a specific personality theory attached (Eysenck - one I have serious problems with).

NB. I haven't used any of the inventories for a long while so I may be out of date.  However, a problem with any of the inventories is that they may not be normed to your population of interest. 

A lie scale essentially involves asking several questions on the same construct in different ways.  If there is significant variation in a person's response to a construct you may wish to analyse the reasons for such variations.  However, self-presentation may also affect all items in the construct equally.

Another approach is to look for external data on a construct and then weight your responses so that they match the distribution you can see in the external data.  If the distribution of responses is skewed in a particular direction (e.g. "our organisation is among the best at …")  one can also apply weightings to the data. 

One option for obtaining other data on the questions of interest, is to use elicitation techniques. I will describe elicitation techniques below.

As an aside,  I should say that I don't actually see the responses as being biased.  'Bias' implies some 'true' value that the questions would otherwise deliver.  The responses we get are accurate responses to the stimulus presented by the question: but we as researchers often don't understand what that stimulus is or what we are doing when we ask the question. See for example

Schwarz, N & Hippler, HJ 1987, 'What response effects may tell your respondents: informative functions of response alternatives', in H. J. Hippler, N. Schwarz and S. Sudman (eds), Social Information Processing and Survey Methodology, Springer-Verlag, New York.

Krosnick, JA 1999, 'Survey Research', Annual Review of Psychology, vol. 50, pp. 537-67

Unintentional self-presentation is inherent in direct questions, particularly in a questionnaire context. Most questionnaires in social sciences either present the participant with the inference  that they are rational human beings with expertise or experience in the area being investigated; or, in the case of  personality inventories etc.,  present as assessing the individual against some external standard.  What you usually get is the response the individual believes is appropriate for them in that context.  Mostly, such responses are 'true' to their understanding of themselves and the question in the context.  When the questionnaire context is close to the situation being investigated (e.g. voting), then the responses are likely to be close but not the same as observed behaviours (Krosnick 1999; Bellman personal communication).

However, and sorry for the long explanatory discussion, elicitation (also known as projective techniques) techniques reduce self-presentation.  For a description of elicitation techniques see

Catterall, M & Ibbotson, P 2000, 'Using Projective Techniques in Education Research', British Educational Research Journal, no. 2, p. 245.

Preliminary interviews using elicitation techniques, or even at the start of a questionnaire, can give you a baseline with relatively low levels of self-presentation that can be used to identify the presence of self-presentation in questionnaires. I have not seen this being done, but in theory and depending on how you collect the data, it can also be used to weight the questionnaire responses.

Conference Paper Understanding through Uncertainty