How Late Can I Pray Fajr?
Question

What is the ruling on delaying Fajr prayer until the sun has almost risen and not offering it at the proper time?
Praise be to Allah.Times of the prayers
The five daily prayers have set times , the beginning and end of which are well defined, and the prayer must be offered on time. Allah says (interpretation of the meaning): 
“Verily, As-Salah (the prayer) is enjoined on the believers at fixed hours” [al-Nisa 4:103]
“This indicates that the prayers are obligatory and have set times, and are not valid at other times. 
These times are well established among the Muslims, young and old, knowledgeable and ignorant, and they learned that from their Prophet Muhammad (peace and blessings of Allah be upon him) who said: “Pray as you have seen me praying.”  (Tafsir al-Sa’di, p. 204)
The ruling on delaying the prayer beyond its proper time
Delaying the prayer beyond its proper time with no excuse is a major sin, and Allah has warned us against it as He says (interpretation of the meaning): 
“So woe unto those performers of Salah (prayers) (hypocrites), Those who delay their Salah (prayer from their stated fixed times)” [al-Ma'un 107:4-5]
“‘Abd-Allah ibn ‘Abbas (may Allah be pleased with him) said: (This means) those who delay their prayers from the proper time.” (Tafsir al-Qurtubi, 20/211)  
Time for Fajr prayer
We have stated in the answer to question no. 9940  the times of the five daily prayers. 
The time for Fajr prayer lasts from the onset of the true dawn until the sun starts to rise. 
The Prophet (peace and blessings of Allah be upon him) said: “The time for Fajr prayer is from the breaking of dawn, so long as the sun has not started to rise.” (Narrated by Muslim, 612)
If a person offers this prayer during this time, he has prayed on time. Based on this, what is mentioned in the question about praying close to sunrise not being the right time is not correct. Rather the time for Fajr prayer lasts until the sun begins to rise. 
The questioner may be referring to some people who delay Fajr prayer until they are certain or think it most likely that the time for it has started. This is due to what is said about mistakes about the time of Fajr in the timetables that are currently available. 
But this mistake is not so serious, rather some scholars have stated that it is only a matter of between 20 and 30 minutes. 
When did the Prophet offer Fajr prayer?
It was the Prophet’s way to pray Fajr before the light of day appeared. This is indicated by a number of ahadith: 
1 – Al-Bukhari (560) and Muslim (646) narrated that Jabir ibn ‘Abd-Allah said: The Prophet (peace and blessings of Allah be upon him) used to pray Fajr when it was still dark (ghalas). 
2- Al-Bukhari (872) and Muslim (645) narrated from ‘Aishah (may Allah be pleased with her) that the Messenger of Allah (peace and blessings of Allah be upon him) used to pray Fajr when it was still dark (ghalas) and the believing women would depart and no one would recognize them because it was so dark, or they would not recognize one another. 
“The word “ghalas'' refers to the darkness at the end of the night, as it says in the dictionary, and that is the beginning of the dawn.” (Subul al-Salam)
Al-Nawawi said: 
“The words “and no one would recognize them because it was so dark” refer to the fact that it was still nighttime and dark. Al-Dawudi said: What this means is that no one could tell if they were women or men.”  (Sharh Muslim by al-Nawawi) 
3 – Ibn Majah (671) narrated that Mughهth ibn Sumay said: I prayed Fajr with ‘Abd-Allah ibn al-Zubayr in the darkness at the end of the night just before daybreak, and when he said the tasleem I turned to Ibn ‘Umar and said, What is this prayer? He said, This is how we prayed with the Messenger of Allah (peace and blessings of Allah be upon him) and with Abu Bakr and ‘Umar. When ‘Umar was stabbed, ‘Uthman delayed it until there was light. Classed as sahih by al-Albani in Sahih Ibn Majah. 
These ahadith indicate that the Prophet (peace and blessings of Allah be upon him) used to pray Fajr at the beginning of its time. 
Ibn Qudamah said in al-Mughni (1/540): 
“With regard to Fajr prayer, it is better to pray when it is still dark. This is the view of Malik, al-Shafi'i and Ishaq. There is also evidence to this effect narrated from Abu Bakr, ‘Umar, Ibn Mas’ud, Abu Musa, Ibn al-Zubayr and ‘Umar ibn ‘Abd al-‘Aziz.”
And Allah knows best.